P2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition.

Size: px
Start display at page:

Download "P2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition."

Transcription

1 P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

2 Jorion, Chapter 7: Portfolio Risk: Analytical Methods DEFINE, CALCULATE, AND DISTINGUISH BETWEEN THE FOLLOWING PORTFOLIO VAR MEASURES: INDIVIDUAL VAR, INCREMENTAL VAR, MARGINAL VAR, COMPONENT VAR, UNDIVERSIFIED PORTFOLIO VAR, AND DIVERSIFIED PORTFOLIO VAR. EXPLAIN THE ROLE OF CORRELATION ON PORTFOLIO RISK

3 Jorion, Chapter 7: Portfolio Risk: Analytical Methods Define, calculate, and distinguish between the following portfolio VaR measures: individual VaR, incremental VaR, marginal VaR, component VaR, undiversified portfolio VaR, and diversified portfolio VaR. Explain the role of correlation on portfolio risk. Describe the challenges associated with VaR measurement as portfolio size increases. Apply the concept of marginal VaR to guide decisions about portfolio VaR. Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio. Explain the difference between risk management and portfolio management, and describe how to use marginal VaR in portfolio management. Define, calculate, and distinguish between the following portfolio VaR measures: individual VaR, incremental VaR, marginal VaR, component VaR, undiversified portfolio VaR, and diversified portfolio VaR. Explain the role of correlation on portfolio risk. Before we elaborate upon the various measures of portfolio VaR, let us try to understand the basic concepts of Portfolio VaR: A portfolio can be characterized by positions on a certain number of constituent assets, expressed in monetary form. In simple terms, portfolio VaR determines the predicted losses with a certain confidence (level) that can be incurred over a certain time horizon. Relationship between Portfolio Returns & Portfolio VaR As the returns of a portfolio can be derived as the linear combination of the returns on underlying assets (assuming that their positions are fixed over the selected horizon) similarly, the VaR of a portfolio can be constructed from a combination of the risks of underlying securities. Therefore, the formula to derive portfolio return is: = + + = Portfolio Return,,, = Weights of each asset in the portfolio containing N number of assets, = Return on each asset. In other words, the expected return of the portfolio, ( ), can be depicted as: ( ) = = 3

4 The variance of the portfolio returns ( ), is derived as: ( ) = = + 2, = Weight of i th asset in the portfolio or = Standard deviation of the returns of i th or j th asset, = Correlation between the returns on asset i and j. We derive the standard deviation of the portfolio returns, of ( ), i.e., by taking the square root = = + 2, Standard deviation is one of the essential factors in deriving VaR as it directly estimates the riskiness of the portfolio. Higher standard deviation may lead to greater losses (or gains). It is important to note that, the correlation plays an important role in defining the magnitude of the calculated VaR. For deriving correlation between each pair of assets in the portfolio, a covariance matrix is used to derive covariance and eventually the correlation coefficient, making it easy to keep track of all covariance terms. To translate the portfolio variance into VaR, we also need to focus upon the distribution of the portfolio return. Consider all individual security returns are normally distributed, such that a linear combination of these normal random variables is also normally distributed. In such a case, we can translate the confidence level c into a standard normal deviate so that the probability of observing losses even more than is c. In other words, is the z-score associated with the confidence level, c. The VaR for a portfolio created with an initial investment of W can be derived as: = W With the derivation of the portfolio VaR, we introduce Individual VaR. Individual VaR It is defined as the VaR of one asset (of a portfolio) measured in isolation. Assuming, w i is the weight of the asset in the portfolio, then individual VaR,, can be derived as: = W = w w i = Proportional weight of the asset in the portfolio W = Total value of the Portfolio The vital point to note here is the usage of the absolute value of position (asset weight). This signifies that the risk exists for both long and short positions and even if the position s weight is negative, the derived risk measure must always be positive. 4

5 Before we explain the remaining Portfolio VaR measures, let us first understand the role of correlation in estimating portfolio risk. Role of Correlation Correlation coefficient helps in deciding the magnitude of the derived VaR. When used intelligently, correlation can play a vital role in reducing the portfolio risk. Portfolio VaR depends upon the number of assets (comprising the portfolio) and the correlation between each pair of those assets. Correlation measures the extent to which two variables (asset returns) move linearly together. If two variables are independent, their correlation is equal to zero. A positive correlation means that the two variables tend to move in the same direction; a negative correlation means that they tend to move in opposite directions. The correlation coefficient always lies between 1 and +1. Also, when the correlation coefficient between the variables is equal to unity then both are said to be perfectly correlated. When 0, the variables are uncorrelated. The correlation coefficient, is derived as: ρ = σ σ σ Where, σ = Covariance between variable 1 & 2 having standard deviation of σ and σ respectively. Reducing Portfolio Risk Lower portfolio risk can be achieved through low correlations or a large number of assets. To see the effect of large number of assets, N, assume that all assets have the same risk and that all correlations are the same, that equal weight is put on each asset. The figure below shows how portfolio risk decreases with the number of assets. As shown in the figure below, the risk of one security is assumed to be 20 percent. When correlation (ρ) is equal to zero, the risk of a 10-asset portfolio drops to 6.3 percent; increasing the assets to 100 drops the risk even further to 2.0 percent. Risk tends asymptotically to zero. More generally, portfolio risk for large number of assets (N) is derived as: = It is evident from the formula above, that the portfolio risk,, tends to zero as N increases. As shown in the figure below, when ρ = 0.5, risk decreases rapidly from 20 to 14.8 percent as N goes to 10 and afterward converges more slowly toward its minimum value of 14.1 percent. Low correlations help to diversify portfolio risk. The above formula also helps in deriving the standard deviation of portfolio for large number of assets. 5

6 Now assess the effect of correlation on portfolio risk in detail. Consider a simple example of a portfolio (P) having two assets, 1 & 2, the diversified portfolio standard deviation is: = ρ And the portfolio variance, VaR P can be derived as: VaR = W = W ρ W = Portfolio Value = z-score associated with the confidence level, c. When the correlation, is zero, the portfolio VaR reduces to (refer Individual VaR): VaR = W + W = VaR + VaR In this case note that the portfolio risk must be lower than the sum of the individual VaRs, VaR < VaR + VaR This reflects the fact that, with the assets that move independently, a portfolio will be less risky than either asset. Thus, VaR is a coherent risk measure for normal and, more generally, elliptical distributions. When the correlation is exactly unity and and are both positive then the portfolio VaR reduces to: VaR = VaR + VaR + 2 VaR VaR = VaR + VaR The portfolio VaR is equal to the sum of the Individual VaR measures if the two assets are perfectly correlated. In general, this will not be the case because correlations typically are imperfect. The benefit from diversification can be measured by the difference between the Diversified VaR and the Undiversified VaR, which is shown in VaR reporting systems. 6

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Tuckman, Chapter 6: Empirical

More information

Brooks, Introductory Econometrics for Finance, 3rd Edition

Brooks, Introductory Econometrics for Finance, 3rd Edition P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Chris Brooks,

More information

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition P2.T5. Market Risk Measurement & Management Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

More information

Kevin Dowd, Measuring Market Risk, 2nd Edition

Kevin Dowd, Measuring Market Risk, 2nd Edition P1.T4. Valuation & Risk Models Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd, Chapter 2: Measures of Financial Risk

More information

FINC 430 TA Session 7 Risk and Return Solutions. Marco Sammon

FINC 430 TA Session 7 Risk and Return Solutions. Marco Sammon FINC 430 TA Session 7 Risk and Return Solutions Marco Sammon Formulas for return and risk The expected return of a portfolio of two risky assets, i and j, is Expected return of asset - the percentage of

More information

P2.T8. Risk Management & Investment Management. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition

P2.T8. Risk Management & Investment Management. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition P2.T8. Risk Management & Investment Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Bodie,

More information

P2.T8. Risk Management & Investment Management. Grinold, Chapter 14: Portfolio Construction

P2.T8. Risk Management & Investment Management. Grinold, Chapter 14: Portfolio Construction P2.T8. Risk Management & Investment Management Grinold, Chapter 14: Portfolio Construction Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Grinold, Chapter 14: Portfolio

More information

P2.T5. Market Risk Measurement & Management

P2.T5. Market Risk Measurement & Management P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Dowd Chapter 3: Estimating

More information

P2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition

P2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd Chapter 3: Estimating Market

More information

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes Reading 40 By David Harper, CFA FRM CIPM www.bionicturtle.com TUCKMAN, CHAPTER

More information

P2.T5. Market Risk Measurement & Management. BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books

P2.T5. Market Risk Measurement & Management. BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books P2.T5. Market Risk Measurement & Management BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books Bionic Turtle FRM Study Notes Reading 38 By David Harper, CFA FRM CIPM

More information

General Notation. Return and Risk: The Capital Asset Pricing Model

General 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 information

Risk and Return and Portfolio Theory

Risk 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 information

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition P2.T5. Market Risk Measurement & Management Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com

More information

Dowd, Measuring Market Risk, 2nd Edition

Dowd, Measuring Market Risk, 2nd Edition P2.T7. Operational & Integrated Risk Management Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes Reading 53 By David Harper, CFA FRM CIPM www.bionicturtle.com DOWD CHAPTER 14: ESTIMATING

More information

Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach

Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach P1.T4. Valuation & Risk Models Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach Bionic Turtle FRM Study Notes Reading 26 By

More information

Hull, Options, Futures, and Other Derivatives, 9 th Edition

Hull, Options, Futures, and Other Derivatives, 9 th Edition P1.T4. Valuation & Risk Models Hull, Options, Futures, and Other Derivatives, 9 th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Sounder www.bionicturtle.com Hull, Chapter

More information

P1.T3. Hull, Chapter 3. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM

P1.T3. Hull, Chapter 3. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM P1.T3. Hull, Chapter 3 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also

More information

John Hull, Risk Management and Financial Institutions, 4th Edition

John Hull, Risk Management and Financial Institutions, 4th Edition P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Chapter 10: Volatility (Learning objectives)

More information

Solutions 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. 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 information

Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach

Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach P1.T3. Financial Markets & Products Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM

More information

Diversification. Finance 100

Diversification. Finance 100 Diversification Finance 100 Prof. Michael R. Roberts 1 Topic Overview How to measure risk and return» Sample risk measures for some classes of securities Brief Statistics Review» Realized and Expected

More information

Port(A,B) is a combination of two stocks, A and B, with standard deviations A and B. A,B = correlation (A,B) = 0.

Port(A,B) is a combination of two stocks, A and B, with standard deviations A and B. A,B = correlation (A,B) = 0. Corporate Finance, Module 6: Risk, Return, and Cost of Capital Practice Problems (The attached PDF file has better formatting.) Updated: July 19, 2007 Exercise 6.1: Minimum Variance Portfolio Port(A,B)

More information

The misleading nature of correlations

The misleading nature of correlations The misleading nature of correlations In this note we explain certain subtle features of calculating correlations between time-series. Correlation is a measure of linear co-movement, to be contrasted with

More information

P2.T6. Credit Risk Measurement & Management. Malz, Financial Risk Management: Models, History & Institutions

P2.T6. Credit Risk Measurement & Management. Malz, Financial Risk Management: Models, History & Institutions P2.T6. Credit Risk Measurement & Management Malz, Financial Risk Management: Models, History & Institutions Portfolio Credit Risk Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Portfolio

More information

Economics 483. Midterm Exam. 1. Consider the following monthly data for Microsoft stock over the period December 1995 through December 1996:

Economics 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 information

Appendix S: Content Portfolios and Diversification

Appendix S: Content Portfolios and Diversification Appendix S: Content Portfolios and Diversification 1188 The expected return on a portfolio is a weighted average of the expected return on the individual id assets; but estimating the risk, or standard

More information

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition P1.T3. Financial Markets & Products Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

More information

P1.T4.Valuation Tuckman, Chapter 5. Bionic Turtle FRM Video Tutorials

P1.T4.Valuation Tuckman, Chapter 5. Bionic Turtle FRM Video Tutorials P1.T4.Valuation Tuckman, Chapter 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal

More information

P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes

P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com BODIE, CHAPTER

More information

Mean-Variance Portfolio Theory

Mean-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 information

Lecture 8 & 9 Risk & Rates of Return

Lecture 8 & 9 Risk & Rates of Return Lecture 8 & 9 Risk & Rates of Return We start from the basic premise that investors LIKE return and DISLIKE risk. Therefore, people will invest in risky assets only if they expect to receive higher returns.

More information

Stulz, Governance, Risk Management and Risk-Taking in Banks

Stulz, Governance, Risk Management and Risk-Taking in Banks P1.T1. Foundations of Risk Stulz, Governance, Risk Management and Risk-Taking in Banks Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Stulz, Governance, Risk Management

More information

Hull, Options, Futures & Other Derivatives

Hull, Options, Futures & Other Derivatives P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Hull, Chapter 1: Introduction

More information

P2.T6. Credit Risk Measurement & Management. Jon Gregory, The xva Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital

P2.T6. Credit Risk Measurement & Management. Jon Gregory, The xva Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital P2.T6. Credit Risk Measurement & Management Jon Gregory, The xva Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM

More information

ERM (Part 1) Measurement and Modeling of Depedencies in Economic Capital. PAK Study Manual

ERM (Part 1) Measurement and Modeling of Depedencies in Economic Capital. PAK Study Manual ERM-101-12 (Part 1) Measurement and Modeling of Depedencies in Economic Capital Related Learning Objectives 2b) Evaluate how risks are correlated, and give examples of risks that are positively correlated

More information

Financial Analysis The Price of Risk. Skema Business School. Portfolio Management 1.

Financial Analysis The Price of Risk. Skema Business School. Portfolio Management 1. Financial Analysis The Price of Risk bertrand.groslambert@skema.edu Skema Business School Portfolio Management Course Outline Introduction (lecture ) Presentation of portfolio management Chap.2,3,5 Introduction

More information

Principles of Finance Risk and Return. Instructor: Xiaomeng Lu

Principles of Finance Risk and Return. Instructor: Xiaomeng Lu Principles of Finance Risk and Return Instructor: Xiaomeng Lu 1 Course Outline Course Introduction Time Value of Money DCF Valuation Security Analysis: Bond, Stock Capital Budgeting (Fundamentals) Portfolio

More information

Portfolio Management

Portfolio Management Portfolio Management Risk & Return Return Income received on an investment (Dividend) plus any change in market price( Capital gain), usually expressed as a percent of the beginning market price of the

More information

Handout 4: Gains from Diversification for 2 Risky Assets Corporate Finance, Sections 001 and 002

Handout 4: Gains from Diversification for 2 Risky Assets Corporate Finance, Sections 001 and 002 Handout 4: Gains from Diversification for 2 Risky Assets Corporate Finance, Sections 001 and 002 Suppose you are deciding how to allocate your wealth between two risky assets. Recall that the expected

More information

Chapter. Diversification and Risky Asset Allocation. McGraw-Hill/Irwin. Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter. Diversification and Risky Asset Allocation. McGraw-Hill/Irwin. Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Diversification and Risky Asset Allocation McGraw-Hill/Irwin Copyright 008 by The McGraw-Hill Companies, Inc. All rights reserved. Diversification Intuitively, we all know that if you hold many

More information

Lecture 3: Return vs Risk: Mean-Variance Analysis

Lecture 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 information

8. International Financial Allocation

8. International Financial Allocation 8. International Financial Allocation An Example and Definitions... 1 Expected eturn, Variance, and Standard Deviation.... S&P 500 Example... The S&P 500 and Treasury bill Portfolio... 8.S. 10-Year Note

More information

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition P1.T3. Financial Markets & Products Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com

More information

ECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 Portfolio Allocation Mean-Variance Approach

ECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 Portfolio Allocation Mean-Variance Approach ECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 ortfolio Allocation Mean-Variance Approach Validity of the Mean-Variance Approach Constant absolute risk aversion (CARA): u(w ) = exp(

More information

Theoretical Aspects Concerning the Use of the Markowitz Model in the Management of Financial Instruments Portfolios

Theoretical Aspects Concerning the Use of the Markowitz Model in the Management of Financial Instruments Portfolios Theoretical Aspects Concerning the Use of the Markowitz Model in the Management of Financial Instruments Portfolios Lecturer Mădălina - Gabriela ANGHEL, PhD Student madalinagabriela_anghel@yahoo.com Artifex

More information

Chapter 10. Chapter 10 Topics. What is Risk? The big picture. Introduction to Risk, Return, and the Opportunity Cost of Capital

Chapter 10. Chapter 10 Topics. What is Risk? The big picture. Introduction to Risk, Return, and the Opportunity Cost of Capital 1 Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital Chapter 10 Topics Risk: The Big Picture Rates of Return Risk Premiums Expected Return Stand Alone Risk Portfolio Return and

More information

Attilio Meucci. Managing Diversification

Attilio Meucci. Managing Diversification Attilio Meucci Managing Diversification A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES

More information

Chapter 8. Markowitz Portfolio Theory. 8.1 Expected Returns and Covariance

Chapter 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 information

Risk and Return: From Securities to Portfolios

Risk and Return: From Securities to Portfolios FIN 614 Risk and Return 2: Portfolios Professor Robert B.H. Hauswald Kogod School of Business, AU Risk and Return: From Securities to Portfolios From securities individual risk and return characteristics

More information

Chapter 5. Asset Allocation - 1. Modern Portfolio Concepts

Chapter 5. Asset Allocation - 1. Modern Portfolio Concepts Asset Allocation - 1 Asset Allocation: Portfolio choice among broad investment classes. Chapter 5 Modern Portfolio Concepts Asset Allocation between risky and risk-free assets Asset Allocation with Two

More information

Answers to Concepts in Review

Answers to Concepts in Review Answers to Concepts in Review 1. A portfolio is simply a collection of investment vehicles assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest expected

More information

... possibly the most important and least understood topic in finance

... possibly the most important and least understood topic in finance Correlation...... possibly the most important and least understood topic in finance 2017 Gary R. Evans. This lecture is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

More information

The mean-variance portfolio choice framework and its generalizations

The 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 information

Archana Khetan 05/09/ MAFA (CA Final) - Portfolio Management

Archana Khetan 05/09/ MAFA (CA Final) - Portfolio Management Archana Khetan 05/09/2010 +91-9930812722 Archana090@hotmail.com MAFA (CA Final) - Portfolio Management 1 Portfolio Management Portfolio is a collection of assets. By investing in a portfolio or combination

More information

SDMR 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) 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 information

Lecture 5. Return and Risk: The Capital Asset Pricing Model

Lecture 5. Return and Risk: The Capital Asset Pricing Model Lecture 5 Return and Risk: The Capital Asset Pricing Model Outline 1 Individual Securities 2 Expected Return, Variance, and Covariance 3 The Return and Risk for Portfolios 4 The Efficient Set for Two Assets

More information

Week 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals

Week 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals Week 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg :

More information

P2.T5. Market Risk Measurement & Management. Hull, Options, Futures, and Other Derivatives, 9th Edition.

P2.T5. Market Risk Measurement & Management. Hull, Options, Futures, and Other Derivatives, 9th Edition. P2.T5. Market Risk Measurement & Management Hull, Options, Futures, and Other Derivatives, 9th Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Hull, Chapter 9:

More information

Derivation Of The Capital Asset Pricing Model Part I - A Single Source Of Uncertainty

Derivation Of The Capital Asset Pricing Model Part I - A Single Source Of Uncertainty Derivation Of The Capital Asset Pricing Model Part I - A Single Source Of Uncertainty Gary Schurman MB, CFA August, 2012 The Capital Asset Pricing Model CAPM is used to estimate the required rate of return

More information

Chapter 6 Efficient Diversification. b. Calculation of mean return and variance for the stock fund: (A) (B) (C) (D) (E) (F) (G)

Chapter 6 Efficient Diversification. b. Calculation of mean return and variance for the stock fund: (A) (B) (C) (D) (E) (F) (G) Chapter 6 Efficient Diversification 1. E(r P ) = 12.1% 3. a. The mean return should be equal to the value computed in the spreadsheet. The fund's return is 3% lower in a recession, but 3% higher in a boom.

More information

Advanced Financial Economics Homework 2 Due on April 14th before class

Advanced 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 information

MS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory

MS-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 information

Portfolio Theory and Diversification

Portfolio Theory and Diversification Topic 3 Portfolio Theoryand Diversification LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain the concept of portfolio formation;. Discuss the idea of diversification; 3. Calculate

More information

P2.T6. Credit Risk Measurement & Management. Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing, Validating and Using Internal Ratings

P2.T6. Credit Risk Measurement & Management. Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing, Validating and Using Internal Ratings P2.T6. Credit Risk Measurement & Management Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing, Validating and Using Internal Ratings Bionic Turtle FRM Practice Questions By David Harper,

More information

Lecture 4: Return vs Risk: Mean-Variance Analysis

Lecture 4: Return vs Risk: Mean-Variance Analysis Lecture 4: Return vs Risk: Mean-Variance Analysis 4.1 Basics Given a cool of many different stocks, you want to decide, for each stock in the pool, whether you include it in your portfolio and (if yes)

More information

P2.T7. Operational & Integrated Risk Management. Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition

P2.T7. Operational & Integrated Risk Management. Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition P2.T7. Operational & Integrated Risk Management Bionic Turtle FRM Practice Questions Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition By David Harper, CFA FRM CIPM

More information

Session 8: The Markowitz problem p. 1

Session 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 information

Chapter 7 1. Random Variables

Chapter 7 1. Random Variables Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous

More information

Finance 100: Corporate Finance. Professor Michael R. Roberts Quiz 3 November 8, 2006

Finance 100: Corporate Finance. Professor Michael R. Roberts Quiz 3 November 8, 2006 Finance 100: Corporate Finance Professor Michael R. Roberts Quiz 3 November 8, 006 Name: Solutions Section ( Points...no joke!): Question Maximum Student Score 1 30 5 3 5 4 0 Total 100 Instructions: Please

More information

Essential Performance Metrics to Evaluate and Interpret Investment Returns. Wealth Management Services

Essential Performance Metrics to Evaluate and Interpret Investment Returns. Wealth Management Services Essential Performance Metrics to Evaluate and Interpret Investment Returns Wealth Management Services Alpha, beta, Sharpe ratio: these metrics are ubiquitous tools of the investment community. Used correctly,

More information

Copyright 2009 Pearson Education Canada

Copyright 2009 Pearson Education Canada Operating Cash Flows: Sales $682,500 $771,750 $868,219 $972,405 $957,211 less expenses $477,750 $540,225 $607,753 $680,684 $670,048 Difference $204,750 $231,525 $260,466 $291,722 $287,163 After-tax (1

More information

Chapter 11. Return and Risk: The Capital Asset Pricing Model (CAPM) Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 11. Return and Risk: The Capital Asset Pricing Model (CAPM) Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 11 Return and Risk: The Capital Asset Pricing Model (CAPM) McGraw-Hill/Irwin Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. 11-0 Know how to calculate expected returns Know

More information

Models of Asset Pricing

Models of Asset Pricing appendix1 to chapter 5 Models of Asset Pricing In Chapter 4, we saw that the return on an asset (such as a bond) measures how much we gain from holding that asset. When we make a decision to buy an asset,

More information

Question # 4 of 15 ( Start time: 07:07:31 PM )

Question # 4 of 15 ( Start time: 07:07:31 PM ) MGT 201 - Financial Management (Quiz # 5) 400+ Quizzes solved by Muhammad Afaaq Afaaq_tariq@yahoo.com Date Monday 31st January and Tuesday 1st February 2011 Question # 1 of 15 ( Start time: 07:04:34 PM

More information

u (x) < 0. and if you believe in diminishing return of the wealth, then you would require

u (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 information

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

More information

An investment s return is your reward for investing. An investment s risk is the uncertainty of what will happen with your investment dollar.

An investment s return is your reward for investing. An investment s risk is the uncertainty of what will happen with your investment dollar. Chapter 7 An investment s return is your reward for investing. An investment s risk is the uncertainty of what will happen with your investment dollar. The relationship between risk and return is a tradeoff.

More information

CHAPTER 8: INDEX MODELS

CHAPTER 8: INDEX MODELS Chapter 8 - Index odels CHATER 8: INDEX ODELS ROBLE SETS 1. The advantage of the index model, compared to the arkowitz procedure, is the vastly reduced number of estimates required. In addition, the large

More information

CHAPTER II LITERATURE STUDY

CHAPTER II LITERATURE STUDY CHAPTER II LITERATURE STUDY 2.1. Risk Management Monetary crisis that strike Indonesia during 1998 and 1999 has caused bad impact to numerous government s and commercial s bank. Most of those banks eventually

More information

Operational Risk Quantification and Insurance

Operational Risk Quantification and Insurance Operational Risk Quantification and Insurance Capital Allocation for Operational Risk 14 th -16 th November 2001 Bahram Mirzai, Swiss Re Swiss Re FSBG Outline Capital Calculation along the Loss Curve Hierarchy

More information

1/12/2011. Chapter 5: z-scores: Location of Scores and Standardized Distributions. Introduction to z-scores. Introduction to z-scores cont.

1/12/2011. Chapter 5: z-scores: Location of Scores and Standardized Distributions. Introduction to z-scores. Introduction to z-scores cont. Chapter 5: z-scores: Location of Scores and Standardized Distributions Introduction to z-scores In the previous two chapters, we introduced the concepts of the mean and the standard deviation as methods

More information

Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk

Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk P1.T4. Valuation & Risk Models Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk Bionic Turtle FRM Study Notes Reading 33 By David Harper, CFA FRM CIPM www.bionicturtle.com DE

More information

The Fallacy of Large Numbers and A Defense of Diversified Active Managers

The Fallacy of Large Numbers and A Defense of Diversified Active Managers The Fallacy of Large umbers and A Defense of Diversified Active Managers Philip H. Dybvig Washington University in Saint Louis First Draft: March 0, 2003 This Draft: March 27, 2003 ABSTRACT Traditional

More information

Module 6 Portfolio risk and return

Module 6 Portfolio risk and return Module 6 Portfolio risk and return Prepared by Pamela Peterson Drake, Ph.D., CFA 1. Overview Security analysts and portfolio managers are concerned about an investment s return, its risk, and whether it

More information

CHAPTER 6: PORTFOLIO SELECTION

CHAPTER 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 information

Bloomberg. Portfolio Value-at-Risk. Sridhar Gollamudi & Bryan Weber. September 22, Version 1.0

Bloomberg. Portfolio Value-at-Risk. Sridhar Gollamudi & Bryan Weber. September 22, Version 1.0 Portfolio Value-at-Risk Sridhar Gollamudi & Bryan Weber September 22, 2011 Version 1.0 Table of Contents 1 Portfolio Value-at-Risk 2 2 Fundamental Factor Models 3 3 Valuation methodology 5 3.1 Linear factor

More information

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology FE670 Algorithmic Trading Strategies Lecture 4. Cross-Sectional Models and Trading Strategies Steve Yang Stevens Institute of Technology 09/26/2013 Outline 1 Cross-Sectional Methods for Evaluation of Factor

More information

Correlation: Its Role in Portfolio Performance and TSR Payout

Correlation: Its Role in Portfolio Performance and TSR Payout Correlation: Its Role in Portfolio Performance and TSR Payout An Important Question By J. Gregory Vermeychuk, Ph.D., CAIA A question often raised by our Total Shareholder Return (TSR) valuation clients

More information

PowerPoint. to accompany. Chapter 11. Systematic Risk and the Equity Risk Premium

PowerPoint. to accompany. Chapter 11. Systematic Risk and the Equity Risk Premium PowerPoint to accompany Chapter 11 Systematic Risk and the Equity Risk Premium 11.1 The Expected Return of a Portfolio While for large portfolios investors should expect to experience higher returns for

More information

Adjusting discount rate for Uncertainty

Adjusting discount rate for Uncertainty Page 1 Adjusting discount rate for Uncertainty The Issue A simple approach: WACC Weighted average Cost of Capital A better approach: CAPM Capital Asset Pricing Model Massachusetts Institute of Technology

More information

Improving Returns-Based Style Analysis

Improving 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 information

All In One MGT201 Mid Term Papers More Than (10) BY

All In One MGT201 Mid Term Papers More Than (10) BY All In One MGT201 Mid Term Papers More Than (10) BY http://www.vustudents.net MIDTERM EXAMINATION MGT201- Financial Management (Session - 2) Question No: 1 ( Marks: 1 ) - Please choose one Why companies

More information

Portfolios of Agricultural Market Advisory Services: How Much Diversification is Enough?

Portfolios of Agricultural Market Advisory Services: How Much Diversification is Enough? Portfolios of Agricultural Market Advisory Services: How Much Diversification is Enough? by Brian G. Stark, Silvina M. Cabrini, Scott H. Irwin, Darrel L. Good, and Joao Martines-Filho Portfolios of Agricultural

More information

Random Variables and Applications OPRE 6301

Random Variables and Applications OPRE 6301 Random Variables and Applications OPRE 6301 Random Variables... As noted earlier, variability is omnipresent in the business world. To model variability probabilistically, we need the concept of a random

More information

Risk Reduction Potential

Risk 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 information

General Disclosure Statement for Transactions

General Disclosure Statement for Transactions I. INTRODUCTION International Swaps and Derivatives Association, Inc. General Disclosure Statement for Transactions We are providing you with this General Disclosure Statement for Transactions ( General

More information

P2.T6. Credit Risk Measurement & Management. Ashcraft & Schuermann, Understanding the Securitization of Subprime Mortgage Credit

P2.T6. Credit Risk Measurement & Management. Ashcraft & Schuermann, Understanding the Securitization of Subprime Mortgage Credit P2.T6. Credit Risk Measurement & Management Ashcraft & Schuermann, Understanding the Securitization of Subprime Mortgage Credit Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa

More information

The Fallacy of Large Numbers

The Fallacy of Large Numbers The Fallacy of Large umbers Philip H. Dybvig Washington University in Saint Louis First Draft: March 0, 2003 This Draft: ovember 6, 2003 ABSTRACT Traditional mean-variance calculations tell us that the

More information

Leverage Aversion, Efficient Frontiers, and the Efficient Region*

Leverage Aversion, Efficient Frontiers, and the Efficient Region* Posted SSRN 08/31/01 Last Revised 10/15/01 Leverage Aversion, Efficient Frontiers, and the Efficient Region* Bruce I. Jacobs and Kenneth N. Levy * Previously entitled Leverage Aversion and Portfolio Optimality:

More information