Value-at-Risk Based Portfolio Management in Electric Power Sector
|
|
- Anna Eaton
- 5 years ago
- Views:
Transcription
1 Value-at-Risk Based Portfolio Management in Electric Power Sector Ran SHI, Jin ZHONG Department of Electrical and Electronic Engineering University of Hong Kong, HKSAR, China ABSTRACT In the deregulated electricity market, highly volatile electricity price leads to the large skew and kurtosis in the price return distribution. Risk management is essential in the portfolio management for market participants. In this paper, we proposed a portfolio optimization model which is capable of managing non-normality in the electricity portfolio return distribution. The optimal portfolio is found by maximizing a performance index resembling to the Sharpe ratio, whereas the risk is defined using Valueat-Risk technique instead of standard deviation. In this model, we can automatically determine the risk-free asset allocation without using utility function, and, as a result, determine how much Contract for Difference(CfD) should be included in the portfolio. After some explanations of the market structure and portfolio elements in Nordic power market, we give an example of a hypothetic electricity company in Oslo, managing its portfolio following the proposed strategy. Keywords: Porfolio Management, Risk Measure, Nonnormality, Value-at-Risk, Nordic Power Market, Risk- Averseness 1. INTRODUCTION Deregulation in electric power sector makes electricity spot prices highly volatile hence induce substantial risk for utilities and power trading companies[1, 2]. In the Nordic electricity market, forward contract hedged by Contract for Difference(CfD) could be considered as riskfree investment in a short term, assuming fuel costs are known when the portfolio managers are making the decision. The risky transactions include forwards, futures, options, CfDs, and other derivatives in both exchange and over-the-counter(otc) market [3]. Further more, the company could also own or lease generation assets in order to get involved in spot trading which is also risky. Hence, how to optimize the portfolio in certain time horizon deserves more treatments. One initiative is applying traditional mean variance approach in finance to electricity market[4, 5]. However, the underlying assumption of mean variance framework is the normality of assets returns which is widely known far from satisfactory in electricity business[6]. Also the concept of utility function is not straightforward in the sense that we do not know how to determine risk averseness and the utility function itself does not have explicit economic meaning. Value-at-Risk(VaR) is a commonly used risk measurement, which is the maximum loss value of the investment over a specified period of time at certain confidence level [7]. To our knowledge, VaR based portfolio management [8, 9] application in electricity market is scarce in literatures. Though a VaR constrained approach has been applied to electric power sector in [10], the method uses the utility function [11], and basis risk is not accounted for. A VaR based optimization framework is proposed in this paper. The method does not required utility function, and it is capable of applying any underlying distribution assumptions. The framework is also applied to a hypothetic company at Oslo Norway in Nordic market. The impacts of degree of risk-averseness are illustrated in the paper. 2. RISK MEASURE Intuitively, risk has two essential components: exposure and uncertainty. Risk, then, is the exposure to a proposition of which one is uncertain. Although we can not control the uncertainty, we can usually control the risk according to the degree of risk-averseness. However, the definition of risk depends on the notion of exposure and uncertainty. Portfolio theory first developed by Markowitz use standard deviation of the expected return to measure the uncertainty, while use the utility function to capture the degree of risk-averseness. Figure 1 plots the standard deviation under normal return distribution. We can see that the risk measure here is just the uncertainty represented by standard deviation. However, when the return distribution is far from normality, which might have flat left tail and be asymmetrical, the standard deviation will not be able to capture the asymmetrical uncertainty which is critical to profitability. Meanwhile, it is not easy to determine the risk-averseness in the utility function, since it does not have explicit economic explanation. Value-at-Risk(VaR) has been adopted by banks and financial institutions as the measure for market risk. VaR is defined as the maximum expected loss on an investment over a specified horizon given some confidence level. The definition of VaR can be given mathematically as follows: Given some confidence level α ( 0,), 1 the VaR of the portfolio at the confidence level α is given by the smallest number l such that the probability that the loss L exceeds l is not larger than (1 α)
2 α-var = inf{ l R : P( L > l ) 1 α} Eq. 1 In probabilistic terms VaR is a quantile of the loss distribution. For example, a 95%-VaR of $4 million under normal trading conditions the trader can be 95% confident that a change in the value of its portfolio would not result in a decrease in portfolio value of more than $4 million during 1 day. This is equivalent to saying that there is a 5% confidence level that the value of its portfolio will decrease by $4 million or more during 1 day. A 95% confidence level does not imply a 95% chance of the event happening, the actual probability of the event cannot be determined. The key point to note is that the target confidence level (95% in the above example) is the given parameter here; the output from the calculation ($4 million in the above example) is the maximum loss (the value at risk) at that confidence level. Figure 1 Standard deviation as risk measure The advantages of VaR lie in the fact that it can capture risk-averseness through pre-defined confidence level, and account for the fat tail in underlying distribution, thus cover the uncertainty more accurately. Also, VaR is essentially a monetary amount, which is easily to be perceived by investors. However, VaR itself does not necessarily represent the risk. Investors usually consider risk in a relative sense by comparing the expected return with other reference measure such as risk free return or industry average return. In this paper, the measure of risk is defined in terms of the VaR of portfolio plus the expected return subject to the level of risk. Figure 2 explained this definition graphically. Please also be advised that the portfolio distribution here is set to be non-normal, which can show the strength of incorporating VaR. ϕ (, c p) = W(0) E( r ) + VaR(, c p) Eq. 2 p where p : portfolio allocation W (0) : initial amount for investment c : confidence level (CL) r p : portfolio return VaR(, c p) : estimated Value-at-Risk of total assets The rationale of defining risk as Eq.2 is to capture the downside risk, and in the mean time, when it is pure riskfree asset, ϕ should be equal to zero. It is obviously satisfied in the risk-free case, where the portfolio Valueat-Risk is simply the negative final wealth, making the ϕ zero. In fact, this risk measure collapses to the standard deviation when the portfolio return follows normal distribution, because in this case, it is just a fixed multiple of standard deviation. Figure 2 Definition of risk associated with VaR 3. PORTFOLIO OPTIMIZATION Portfolio optimization is the rational process of seeking optimal risk-return trade off. Basically the trade-off is quantified by sort of yield per unit of risk. Markowitz is the father of modern portfolio theory by proposing meanvariance framework. In this framework, covariance plays an important role in this theory, because unsystematic risk can be diversified away by selecting negatively correlated assets, thus we can construct portfolio with the same expected return but less risk by considering the interactions between assets. In the mean-variance framework, the trade-off is defined by the Sharpe ratio [12] equal to the expected portfolio return in excess of risk-free return, divided by standard deviation of the expected return. This framework can be presented by a two-step approach. This first step is to obtain optimal risky portfolio by maximizing Sharpe ratio, which is the slope of the capital market line (CAL). The second step is to obtain the optimal allocation between risk-free investment and optimal risky portfolio. The two step process is graphically presented as in Figure 3 for the purpose of comparison with VaR-based approach later..
3 variance framework has now become efficient mean ϕ frontier. Since risk-free investment does not affect the allocation of risky investment, we will first obtain the optimal risky portfolio by maximizing S(p). Then, we will show that the total allocation will be automatically determined by the desired VaR without resorting to utility function. Figure 3 Portfolio selection under mean-variance framework We should note that risk preference comes in the second step as a parameter pre-determined in utility function. However, how to define proper utility function becomes another problem, and the solution is usually not clear. Since mean-variance framework relies on the covariance structure, it is only possible to calculate portfolio variance under the assumption that all asset returns follow normal distribution. This brings the major problem of applying mean-variance framework to electricity market because electricity price is highly volatile, with characteristics such as seasonality, mean-reversion, spikes, hence the return in electricity portfolio is far from normality. VaRbased portfolio selection strategy can cope with this very well. The framework is explained as follows. Suppose W (0) is the initial amount of fund to be invested in the horizon T, which we want to invest such that the portfolio meets a chosen VaR limit. This amount could be invested along with amount B representing borrowing (B>0) or lending (B<0) at the risk free rate. Therefore the final wealth is The optimal portfolio is chosen independently from the level of initial amount of fund. It is also independent from the desired VaR, since the risk measure depends on the estimated portfolio VaR rather than the desired VaR. However, as the investor s degree of the risk aversion is already captured by the desired VaR and confidence level, the amount of borrowing or lending required to meet the VaR constraint is therefore determined as following: * W(0)( VaR VaR( c, p')) B = Eq. 5 ϕ '( c, p') We can find that if the portfolio manager set the desired VaR equal to the expected portfolio VaR, no risk-free investment will be involved. In the other way around, if there is no risk free investment exists, the desired VaR could only be the expected portfolio VaR. The whole process is graphically presented in Figure 4 * VaR VaR(, c p ') E0 W T p W B r p B r f ( (, )) = ( (0) + )(1 + ( )) (1 + ) Eq. 3 The Value-at-Risk based portfolio optimization mathematically formulated as follows: is ϕ '( c, p') Figure 4 VaR-based portfolio optimization scheme Where r( p) rf p' = max S( p) = p ϕ ( c, p) S( p) : performance index p ' :optimal portfolio r f : risk free rate of return r( p) : total return on initial wealth Eq. 4 The denominator in Eq.4 is as defined in Eq.2. Obviously the performance index S( p) resembles Sharp Ratio assuming the return distribution is multivariate normality. We will still maximize the trade-off between risk and return. The efficient frontier in the mean- 4. NORDIC POWER MARKET Imagine an integrated utility company which may own or lease generation assets, and also has a trading division that can trade for forward, futures, options, CfDs as well as other exotic options. We refer all potential assets for the portfolio including owned or leased generation assets as instruments. In fact, all these instruments can be conceptualized as option contracts on certain capacities (in MW) which could be called or sold in a specific period. Each instrument has a strike price at which the option is exercised, and an option premium at which the underlying capacity is reserved. In this framework, owned generation assets have strike prices equal to marginal costs, and zero option premium. Leased
4 generation has strike price equal to marginal cost, and leasing rate as option premium. The forward, which is prepaid, is obligated to deliver power, hence it has zero strike price. The settlements of forwards, options are based on Systems Prices. However, the actual power procurement cost is based on area price of whatever area the power is delivered to. System price is some sort of average of area prices. When transmission congestion occurs, area prices will be different because cheaper power cannot be delivered to demand in other areas freely, hence the whole market can not achieve equilibrium where all prices should be overall marginal cost of all generation assets. In fact transmission congestion is not uncommon, thus areas prices differ very often. In this case, forwards become risky in the presence of basis risk, the difference between system price and area price of which the power is going to. CfD is based on the difference between the area prices and system prices, and it is designed to hedge the basis risk. To create a perfect hedge that includes the basis risk when area prices are not equal to the System Prices, a three-step process using CfDs must be followed: Hedge the required volume using forward contract. Hedge any price difference for the same period and volume through CfDs Accomplish physical procurement by trade in the spot market area of the member s location.[3] return is the return of CfD-hedged forward return, which is 41%. Using the empirical data of 9 years, we equivalently obtain 279 scenarios for each random portfolio allocation. The 95%-VaR of each random portfolio allocation is obtained by searching for the lowest 5 percent quantile in the portfolio return distribution random allocations are plot in Figure 5, where the optimal risky allocation is obtained through Eq.4, which is at the tangential point in Figure 5. Table 2 below shows the optimal weights of elements 1-4 in the portfolio setting. The portfolio VaR is million NoK, which means, without involving risk-free assets, the decision maker should have a desired VaR equal to million NoK after the optimal allocation is made. Label Weights Table 2 Optimal weights Now, we will show how much CfD is needed to cater for the decision maker s desired VaR. Suppose the decision maker has a desired 95%-VaR different from million NoK, the risk-free asset could be obtained directly using Eq.5, without resorting to utility function. Thereafter, the CfD portion and additional forwards are obtained since the risk-free asset is constructed as forward contract fully hedged by CfDs. 5. EXAMPLES We construct a company operating at Oslo Norway, using area prices of Oslo and Nordpool system prices from 1998 to All prices used are daily average prices obtained from nordpool website [13]. The company uses spot trading, forwards, CfDs, and options to make profit while meet the load demand. The trading period is August. In this month, the Oslo prices are usually lower than system prices due to the abundant water supply to the hydropower generators in Norway. Assuming the initial money W (0) is one million Norwegian Krone (NoK). In this example, two cases are considered to illustrate the procedure of the proposed portfolio management framework, and demonstrate the advantages of capturing risk-averseness of decision maker respectively. Case 1 Asset Label Strike Price Premium Generation Aug Forward Daily Call Daily Put Aug CfD Table 1 Portfolio elements Here, the CfD is assumed to cover the same capacity of one monthly forward contract. Therefore, the risk-free Figure 5 Efficient frontier and optimal risky allocation Case 2 Varying the desired VaR can help us find the risk-free allocation, which captures a part of risk-averseness of decision maker. Also, varying the confidence level in the VaR setting provides us another way to incorporate riskaverseness. When the decision maker is more conservative, and has a larger confidence level, the portfolio VaR in the risk measure ϕ (, cp) will become larger, thus make the risk larger. Intuitively, if the return distribution is strictly normal, this will not change our optimal allocation, because our defined risk measure at any confidence level is just a multiple of standard
5 deviation. However, it s not the case when we have other leptokurtic distribution assumptions, or simply using the empirical distribution. Figure 6 shows the shift of efficient frontier towards RHS as the confidence level changed from 90% to 99%. Expected Return Expected Return Risk/ Million NoK( 99% CL) [6] R. Weron and I. Simonsen, "Blackouts, risk, and fattailed distributions," Springer Verlag, [7] P. Jorion, Value at risk: the new benchmark for controlling market risk: McGraw-Hill, [8] A. Puelz, "Value-at-Risk Based Portfolio Optimization," [9] R. Campbell, R. Huisman, and K. Koedijk, "Optimal portfolio selection in a Value-at-Risk framework." vol. 25: Elsevier Science, 2001, pp [10] P. R. Kleindorfer and L. Li, "Multi-period VaRconstrained portfolio optimization with applications to the electric power sector." vol. 26, 2005, pp [11] H. M. Markowitz, "Foundations of portfolio theory," World Scientific, [12] W. F. Sharpe, "The Sharpe Ratio," Princeton University Press, [13] Risk/Million NoK ( 90% CL) Figure 6 Impact of varying confidence level 6. CONCLUSIONS The need for capturing downside risk motivates us to incorporate the Value-at-Risk in the risk measure. Since the risk-averseness of decision maker could be well accounted for in both the confidence level and desired VaR level, there is no need to use utility function to find the optimal allocation between risk-free and risky assets. The portfolio optimization criterion is designed as to maximizing the downside risk-return trade-off which is defined as. One special result in electric power sector is that the use of CfD can be determined automatically as the risk-free proportion is found, since the risk-free asset is considered as the forward contract fully hedged by CfDs. Since our framework is capable of incorporating any return distribution assumption, one natural extension of this paper could be the comparison among more leptokurtic distributions assumptions, in additional to empirical distribution and normal distribution. 7. REFERENCES [1] S. J. Deng and S. S. Oren, "Electricity derivatives and risk management." vol. 31: Elsevier, 2006, pp [2] J. J. Lucia and E. S. Schwartz, "Electricity Prices and Power Derivatives: Evidence from the Nordic Power Exchange." vol. 5: Springer, 2002, pp [3] A. S. A. Nord Pool, "Trade at Nord Pool s Financial Market," April, [4] M. Liu and F. F. Wu, "Portfolio optimization in electricity markets." vol. 77: Elsevier, 2007, pp [5] H. M. Markowitz, Mean-variance analysis in portfolio choice and capital markets: Basil Blackwell, 1987.
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 informationAGENERATION company s (Genco s) objective, in a competitive
1512 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006 Managing Price Risk in a Multimarket Environment Min Liu and Felix F. Wu, Fellow, IEEE Abstract In a competitive electricity market,
More informationThe Effect of Widespread Use of Value-at-Risk on Liquidity and Prices in the Nordic Power Market
The Effect of Widespread Use of Value-at-Risk on Liquidity and Prices in the Nordic Power Market Cathrine Pihl Næss Adviser, Nord Pool Spot AS Direct phone: +47 67 52 80 73 Fax: +47 67 52 81 02 E-mail:
More informationLeverage 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 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 informationOptimization of a Real Estate Portfolio with Contingent Portfolio Programming
Mat-2.108 Independent research projects in applied mathematics Optimization of a Real Estate Portfolio with Contingent Portfolio Programming 3 March, 2005 HELSINKI UNIVERSITY OF TECHNOLOGY System Analysis
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 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 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 informationCHAPTER 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 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 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 informationDoes an Optimal Static Policy Foreign Currency Hedge Ratio Exist?
May 2015 Does an Optimal Static Policy Foreign Currency Hedge Ratio Exist? FQ Perspective DORI LEVANONI Partner, Investments Investing in foreign assets comes with the additional question of what to do
More informationCOPYRIGHTED MATERIAL. Portfolio Selection CHAPTER 1. JWPR026-Fabozzi c01 June 22, :54
CHAPTER 1 Portfolio Selection FRANK J. FABOZZI, PhD, CFA, CPA Professor in the Practice of Finance, Yale School of Management HARRY M. MARKOWITZ, PhD Consultant FRANCIS GUPTA, PhD Director, Research, Dow
More informationPORTFOLIO OPTIMIZATION AND SHARPE RATIO BASED ON COPULA APPROACH
VOLUME 6, 01 PORTFOLIO OPTIMIZATION AND SHARPE RATIO BASED ON COPULA APPROACH Mária Bohdalová I, Michal Gregu II Comenius University in Bratislava, Slovakia In this paper we will discuss the allocation
More informationKEIR EDUCATIONAL RESOURCES
INVESTMENT PLANNING 2017 Published by: KEIR EDUCATIONAL RESOURCES 4785 Emerald Way Middletown, OH 45044 1-800-795-5347 1-800-859-5347 FAX E-mail customerservice@keirsuccess.com www.keirsuccess.com TABLE
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 informationSolving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?
DOI 0.007/s064-006-9073-z ORIGINAL PAPER Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function? Jules H. van Binsbergen Michael W. Brandt Received:
More informationAsset Allocation Model with Tail Risk Parity
Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference 2017 Asset Allocation Model with Tail Risk Parity Hirotaka Kato Graduate School of Science and Technology Keio University,
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 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 informationArchana 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 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 informationCovariance Matrix Estimation using an Errors-in-Variables Factor Model with Applications to Portfolio Selection and a Deregulated Electricity Market
Covariance Matrix Estimation using an Errors-in-Variables Factor Model with Applications to Portfolio Selection and a Deregulated Electricity Market Warren R. Scott, Warren B. Powell Sherrerd Hall, Charlton
More informationPortfolio Selection using Data Envelopment Analysis (DEA): A Case of Select Indian Investment Companies
ISSN: 2347-3215 Volume 2 Number 4 (April-2014) pp. 50-55 www.ijcrar.com Portfolio Selection using Data Envelopment Analysis (DEA): A Case of Select Indian Investment Companies Leila Zamani*, Resia Beegam
More informationA Simple Utility Approach to Private Equity Sales
The Journal of Entrepreneurial Finance Volume 8 Issue 1 Spring 2003 Article 7 12-2003 A Simple Utility Approach to Private Equity Sales Robert Dubil San Jose State University Follow this and additional
More informationDoes Portfolio Theory Work During Financial Crises?
Does Portfolio Theory Work During Financial Crises? Harry M. Markowitz, Mark T. Hebner, Mary E. Brunson It is sometimes said that portfolio theory fails during financial crises because: All asset classes
More informationThe Capital Assets Pricing Model & Arbitrage Pricing Theory: Properties and Applications in Jordan
Modern Applied Science; Vol. 12, No. 11; 2018 ISSN 1913-1844E-ISSN 1913-1852 Published by Canadian Center of Science and Education The Capital Assets Pricing Model & Arbitrage Pricing Theory: Properties
More informationMULTISTAGE PORTFOLIO OPTIMIZATION AS A STOCHASTIC OPTIMAL CONTROL PROBLEM
K Y B E R N E T I K A M A N U S C R I P T P R E V I E W MULTISTAGE PORTFOLIO OPTIMIZATION AS A STOCHASTIC OPTIMAL CONTROL PROBLEM Martin Lauko Each portfolio optimization problem is a trade off between
More informationReturn and Risk: The Capital-Asset Pricing Model (CAPM)
Return and Risk: The Capital-Asset Pricing Model (CAPM) Expected Returns (Single assets & Portfolios), Variance, Diversification, Efficient Set, Market Portfolio, and CAPM Expected Returns and Variances
More informationMean-Variance Model for Portfolio Selection
Mean-Variance Model for Portfolio Selection FRANK J. FABOZZI, PhD, CFA, CPA Professor of Finance, EDHEC Business School HARRY M. MARKOWITZ, PhD Consultant PETTER N. KOLM, PhD Director of the Mathematics
More informationPortfolio 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 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 informationJournal of Computational and Applied Mathematics. The mean-absolute deviation portfolio selection problem with interval-valued returns
Journal of Computational and Applied Mathematics 235 (2011) 4149 4157 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
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 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 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 informationMeasuring the Systematic Risk of Stocks Using the Capital Asset Pricing Model
Journal of Investment and Management 2017; 6(1): 13-21 http://www.sciencepublishinggroup.com/j/jim doi: 10.11648/j.jim.20170601.13 ISSN: 2328-7713 (Print); ISSN: 2328-7721 (Online) Measuring the Systematic
More informationAsset Allocation. Cash Flow Matching and Immunization CF matching involves bonds to match future liabilities Immunization involves duration matching
Asset Allocation Strategic Asset Allocation Combines investor s objectives, risk tolerance and constraints with long run capital market expectations to establish asset allocations Create the policy portfolio
More informationThe risk/return trade-off has been a
Efficient Risk/Return Frontiers for Credit Risk HELMUT MAUSSER AND DAN ROSEN HELMUT MAUSSER is a mathematician at Algorithmics Inc. in Toronto, Canada. DAN ROSEN is the director of research at Algorithmics
More informationMSc Finance Birkbeck University of London Theory of Finance I. Lecture Notes
MSc Finance Birkbeck University of London Theory of Finance I Lecture Notes 2006-07 This course introduces ideas and techniques that form the foundations of theory of finance. The first part of the course,
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 informationTHEORY & PRACTICE FOR FUND MANAGERS. SPRING 2011 Volume 20 Number 1 RISK. special section PARITY. The Voices of Influence iijournals.
T H E J O U R N A L O F THEORY & PRACTICE FOR FUND MANAGERS SPRING 0 Volume 0 Number RISK special section PARITY The Voices of Influence iijournals.com Risk Parity and Diversification EDWARD QIAN EDWARD
More informationExecutive Summary: A CVaR Scenario-based Framework For Minimizing Downside Risk In Multi-Asset Class Portfolios
Executive Summary: A CVaR Scenario-based Framework For Minimizing Downside Risk In Multi-Asset Class Portfolios Axioma, Inc. by Kartik Sivaramakrishnan, PhD, and Robert Stamicar, PhD August 2016 In this
More informationPortfolio 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 informationUniversity 18 Lessons Financial Management. Unit 12: Return, Risk and Shareholder Value
University 18 Lessons Financial Management Unit 12: Return, Risk and Shareholder Value Risk and Return Risk and Return Security analysis is built around the idea that investors are concerned with two principal
More informationSession 10: Lessons from the Markowitz framework p. 1
Session 10: Lessons from the Markowitz framework Susan Thomas http://www.igidr.ac.in/ susant susant@mayin.org IGIDR Bombay Session 10: Lessons from the Markowitz framework p. 1 Recap The Markowitz question:
More informationEfficient Frontier and Asset Allocation
Topic 4 Efficient Frontier and Asset Allocation LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain the concept of efficient frontier and Markowitz portfolio theory; 2. Discuss
More informationTraditional Optimization is Not Optimal for Leverage-Averse Investors
Posted SSRN 10/1/2013 Traditional Optimization is Not Optimal for Leverage-Averse Investors Bruce I. Jacobs and Kenneth N. Levy forthcoming The Journal of Portfolio Management, Winter 2014 Bruce I. Jacobs
More informationFNCE 4030 Fall 2012 Roberto Caccia, Ph.D. Midterm_2a (2-Nov-2012) Your name:
Answer the questions in the space below. Written answers require no more than few compact sentences to show you understood and master the concept. Show your work to receive partial credit. Points are as
More informationECMC49S Midterm. Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100
ECMC49S Midterm Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [10 marks] (i) State the Fisher Separation Theorem
More informationExpected shortfall or median shortfall
Journal of Financial Engineering Vol. 1, No. 1 (2014) 1450007 (6 pages) World Scientific Publishing Company DOI: 10.1142/S234576861450007X Expected shortfall or median shortfall Abstract Steven Kou * and
More informationPortfolio Optimization using Conditional Sharpe Ratio
International Letters of Chemistry, Physics and Astronomy Online: 2015-07-01 ISSN: 2299-3843, Vol. 53, pp 130-136 doi:10.18052/www.scipress.com/ilcpa.53.130 2015 SciPress Ltd., Switzerland Portfolio Optimization
More informationKEIR EDUCATIONAL RESOURCES
INVESTMENT PLANNING 2015 Published by: KEIR EDUCATIONAL RESOURCES 4785 Emerald Way Middletown, OH 45044 1-800-795-5347 1-800-859-5347 FAX E-mail customerservice@keirsuccess.com www.keirsuccess.com 2015
More informationA Study on the Risk Regulation of Financial Investment Market Based on Quantitative
80 Journal of Advanced Statistics, Vol. 3, No. 4, December 2018 https://dx.doi.org/10.22606/jas.2018.34004 A Study on the Risk Regulation of Financial Investment Market Based on Quantitative Xinfeng Li
More informationFinancial Market Analysis (FMAx) Module 6
Financial Market Analysis (FMAx) Module 6 Asset Allocation and iversification This training material is the property of the International Monetary Fund (IMF) and is intended for use in IMF Institute for
More informationRisk and Return. Nicole Höhling, Introduction. Definitions. Types of risk and beta
Risk and Return Nicole Höhling, 2009-09-07 Introduction Every decision regarding investments is based on the relationship between risk and return. Generally the return on an investment should be as high
More informationModern Portfolio Theory -Markowitz Model
Modern Portfolio Theory -Markowitz Model Rahul Kumar Project Trainee, IDRBT 3 rd year student Integrated M.Sc. Mathematics & Computing IIT Kharagpur Email: rahulkumar641@gmail.com Project guide: Dr Mahil
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationOMEGA. A New Tool for Financial Analysis
OMEGA A New Tool for Financial Analysis 2 1 0-1 -2-1 0 1 2 3 4 Fund C Sharpe Optimal allocation Fund C and Fund D Fund C is a better bet than the Sharpe optimal combination of Fund C and Fund D for more
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 informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Fall 2017 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationCHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS. McGraw-Hill/Irwin
CHAPTER 5 Introduction to Risk, Return, and the Historical Record McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 5-2 Interest Rate Determinants Supply Households
More informationFinancial Economics 4: Portfolio Theory
Financial Economics 4: Portfolio Theory Stefano Lovo HEC, Paris What is a portfolio? Definition A portfolio is an amount of money invested in a number of financial assets. Example Portfolio A is worth
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Spring 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationCSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization
CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization March 9 16, 2018 1 / 19 The portfolio optimization problem How to best allocate our money to n risky assets S 1,..., S n with
More informationA folk theorem for one-shot Bertrand games
Economics Letters 6 (999) 9 6 A folk theorem for one-shot Bertrand games Michael R. Baye *, John Morgan a, b a Indiana University, Kelley School of Business, 309 East Tenth St., Bloomington, IN 4740-70,
More informationThe 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 informationBudget Setting Strategies for the Company s Divisions
Budget Setting Strategies for the Company s Divisions Menachem Berg Ruud Brekelmans Anja De Waegenaere November 14, 1997 Abstract The paper deals with the issue of budget setting to the divisions of a
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 informationCOMPARISON OF NATURAL HEDGES FROM DIVERSIFICATION AND DERIVATE INSTRUMENTS AGAINST COMMODITY PRICE RISK : A CASE STUDY OF PT ANEKA TAMBANG TBK
THE INDONESIAN JOURNAL OF BUSINESS ADMINISTRATION Vol. 2, No. 13, 2013:1651-1664 COMPARISON OF NATURAL HEDGES FROM DIVERSIFICATION AND DERIVATE INSTRUMENTS AGAINST COMMODITY PRICE RISK : A CASE STUDY OF
More informationOcean Hedge Fund. James Leech Matt Murphy Robbie Silvis
Ocean Hedge Fund James Leech Matt Murphy Robbie Silvis I. Create an Equity Hedge Fund Investment Objectives and Adaptability A. Preface on how the hedge fund plans to adapt to current and future market
More informationPRICING ASPECTS OF FORWARD LOCATIONAL PRICE DIFFERENTIAL PRODUCTS
PRICING ASPECTS OF FORWARD LOCATIONAL PRICE DIFFERENTIAL PRODUCTS Tarjei Kristiansen Norwegian University of Science and Technology and Norsk Hydro ASA Oslo, Norway Tarjei.Kristiansen@elkraft.ntnu.no Abstract
More information23.1. Assumptions of Capital Market Theory
NPTEL Course Course Title: Security Analysis and Portfolio anagement Course Coordinator: Dr. Jitendra ahakud odule-12 Session-23 Capital arket Theory-I Capital market theory extends portfolio theory and
More informationApplying Index Investing Strategies: Optimising Risk-adjusted Returns
Applying Index Investing Strategies: Optimising -adjusted Returns By Daniel R Wessels July 2005 Available at: www.indexinvestor.co.za For the untrained eye the ensuing topic might appear highly theoretical,
More informationEquation Chapter 1 Section 1 A Primer on Quantitative Risk Measures
Equation Chapter 1 Section 1 A rimer on Quantitative Risk Measures aul D. Kaplan, h.d., CFA Quantitative Research Director Morningstar Europe, Ltd. London, UK 25 April 2011 Ever since Harry Markowitz s
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 informationPortfolio Management
Portfolio Management 010-011 1. Consider the following prices (calculated under the assumption of absence of arbitrage) corresponding to three sets of options on the Dow Jones index. Each point of the
More informationMEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL
MEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL Isariya Suttakulpiboon MSc in Risk Management and Insurance Georgia State University, 30303 Atlanta, Georgia Email: suttakul.i@gmail.com,
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 informationLecture 6: Non Normal Distributions
Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015 Overview Non-normalities in (standardized) residuals from asset return
More informationValue at Risk, Expected Shortfall, and Marginal Risk Contribution, in: Szego, G. (ed.): Risk Measures for the 21st Century, p , Wiley 2004.
Rau-Bredow, Hans: Value at Risk, Expected Shortfall, and Marginal Risk Contribution, in: Szego, G. (ed.): Risk Measures for the 21st Century, p. 61-68, Wiley 2004. Copyright geschützt 5 Value-at-Risk,
More informationLiquidity Creation as Volatility Risk
Liquidity Creation as Volatility Risk Itamar Drechsler, NYU and NBER Alan Moreira, Rochester Alexi Savov, NYU and NBER JHU Carey Finance Conference June, 2018 1 Liquidity and Volatility 1. Liquidity creation
More informationA Comparative Research on Banking Sector and Performance Between China and Pakistan (National Bank of Pakistan Versus Agricultural Bank of China)
American Journal of Economics, Finance and Management Vol. 1, No. 6, 2015, pp. 594-598 http://www.aiscience.org/journal/ajefm ISSN: 2381-6864 (Print); ISSN: 2381-6902 (Online) A Comparative Research on
More informationAn Empirical Examination of the Electric Utilities Industry. December 19, Regulatory Induced Risk Aversion in. Contracting Behavior
An Empirical Examination of the Electric Utilities Industry December 19, 2011 The Puzzle Why do price-regulated firms purchase input coal through both contract Figure and 1(a): spot Contract transactions,
More 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 informationFINC 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 informationSample Midterm Questions Foundations of Financial Markets Prof. Lasse H. Pedersen
Sample Midterm Questions Foundations of Financial Markets Prof. Lasse H. Pedersen 1. Security A has a higher equilibrium price volatility than security B. Assuming all else is equal, the equilibrium bid-ask
More informationMeasuring Financial Risk using Extreme Value Theory: evidence from Pakistan
Measuring Financial Risk using Extreme Value Theory: evidence from Pakistan Dr. Abdul Qayyum and Faisal Nawaz Abstract The purpose of the paper is to show some methods of extreme value theory through analysis
More informationAPPLYING MULTIVARIATE
Swiss Society for Financial Market Research (pp. 201 211) MOMTCHIL POJARLIEV AND WOLFGANG POLASEK APPLYING MULTIVARIATE TIME SERIES FORECASTS FOR ACTIVE PORTFOLIO MANAGEMENT Momtchil Pojarliev, INVESCO
More informationCHAPTER III RISK MANAGEMENT
CHAPTER III RISK MANAGEMENT Concept of Risk Risk is the quantified amount which arises due to the likelihood of the occurrence of a future outcome which one does not expect to happen. If one is participating
More informationHo Ho Quantitative Portfolio Manager, CalPERS
Portfolio Construction and Risk Management under Non-Normality Fiduciary Investors Symposium, Beijing - China October 23 rd 26 th, 2011 Ho Ho Quantitative Portfolio Manager, CalPERS The views expressed
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 informationBanking firm and hedging over the business cycle. Citation Portuguese Economic Journal, 2010, v. 9 n. 1, p
Title Banking firm and hedging over the business cycle Author(s) Broll, U; Wong, KP Citation Portuguese Economic Journal, 2010, v. 9 n. 1, p. 29-33 Issued Date 2010 URL http://hdl.handle.net/10722/124052
More informationYale ICF Working Paper No First Draft: February 21, 1992 This Draft: June 29, Safety First Portfolio Insurance
Yale ICF Working Paper No. 08 11 First Draft: February 21, 1992 This Draft: June 29, 1992 Safety First Portfolio Insurance William N. Goetzmann, International Center for Finance, Yale School of Management,
More informationFinancial Economics Field Exam August 2011
Financial Economics Field Exam August 2011 There are two questions on the exam, representing Macroeconomic Finance (234A) and Corporate Finance (234C). Please answer both questions to the best of your
More informationComparison of Estimation For Conditional Value at Risk
-1- University of Piraeus Department of Banking and Financial Management Postgraduate Program in Banking and Financial Management Comparison of Estimation For Conditional Value at Risk Georgantza Georgia
More informationMotif Capital Horizon Models: A robust asset allocation framework
Motif Capital Horizon Models: A robust asset allocation framework Executive Summary By some estimates, over 93% of the variation in a portfolio s returns can be attributed to the allocation to broad asset
More informationConsumption and Portfolio Choice under Uncertainty
Chapter 8 Consumption and Portfolio Choice under Uncertainty In this chapter we examine dynamic models of consumer choice under uncertainty. We continue, as in the Ramsey model, to take the decision of
More informationPortfolio Optimization. Prof. Daniel P. Palomar
Portfolio Optimization Prof. Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) MAFS6010R- Portfolio Optimization with R MSc in Financial Mathematics Fall 2018-19, HKUST, Hong
More information