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Portfolio Optimization with Drawdown Constraints
Implemented in Portfolio Safeguard by AOrDa.com PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS Alexei Chekhlov1, Stanislav Uryasev2, Michael Zabarankin2 Risk Management and Financial Engineering Lab Center for Applied Optimization Department of Industrial and Systems Engineering University of Florida, Gainesville, FL 32611 Date: January 13, 2003 Correspondence should be addressed to: Stanislav Uryasev Abstract We propose a new one-parameter family of risk functions defined on portfolio return sample -paths, which is called conditional drawdown-at-risk (CDaR). These risk functions depend on the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter a , the CDaR is defined as the mean of the worst (1-a) *100% drawdowns. The CDaR risk function contains the maximal drawdown and average drawdown as its limiting cases. For a particular example, we find optimal portfolios with constraints on the maximal drawdown, average drawdown and several intermediate cases between these two. The CDaR family of risk functions originates from the conditional value-at-risk (CVaR) measure. Some recommendations on how to select the optimal risk measure for getting practically stable portfolios are provided. We solved a real life portfolio allocation problem using the proposed risk functions. 1. Introduction Optimal portfolio allocation is a longstanding issue in both practical portfolio management and academic research on portfolio theory. Various methods have been proposed and studied (for a review, see, for example, Grinold and Kahn, 1999). All of them, as a starting point, assume some measure of portfolio risk. From a standpoint of a fund manager, who trades clients' or bank’s proprietary capital, and for whom the clients' accounts are the only source of income coming in the form of management and incentive fees, losing these accounts is equivalent to the death of his business. -
ESG Considerations for Securitized Fixed Income Notes
ESG Considerations for Securitized Fixed Neil Hohmann, PhD Managing Director Income Notes Head of Structured Products BBH Investment Management [email protected] +1.212.493.8017 Executive Summary Securitized assets make up over a quarter of the U.S. corporations since 2010, we find median price declines fixed income markets,1 yet assessments of environmental, for equity of those companies of -16%, and -3% median social, and governance (ESG) risks related to this sizable declines for their corporate bonds.3 Yet, the median price segment of the bond market are notably lacking. decline for securitized notes during these periods is 0% – securitized notes are as likely to climb in price as they are The purpose of this study is to bring securitized notes to fall in price, through these incidents. This should not be squarely into the realm of responsible investing through the surprising – securitizations are designed to insulate inves- development of a specialized ESG evaluation framework tors from corporate distress. They have a senior security for securitizations. To date, securitization has been notably interest in collateral, ring-fenced legal structures and absent from responsible investing discussions, probably further structural protections – all of which limit their owing to the variety of securitization types, their structural linkage to the originating company. However, we find that complexities and limited knowledge of the sector. Manag- a few securitization asset types, including whole business ers seem to either exclude securitizations from ESG securitizations and RMBS, can still bear substantial ESG assessment or lump them into their corporate exposures. risk. A specialized framework for securitizations is needed. -
Portfolio Optimization with Drawdown Constraints
PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS Alexei Chekhlov1, Stanislav Uryasev2, Michael Zabarankin2 Risk Management and Financial Engineering Lab Center for Applied Optimization Department of Industrial and Systems Engineering University of Florida, Gainesville, FL 32611 Date: January 13, 2003 Correspondence should be addressed to: Stanislav Uryasev Abstract We propose a new one-parameter family of risk functions defined on portfolio return sample -paths, which is called conditional drawdown-at-risk (CDaR). These risk functions depend on the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter a , the CDaR is defined as the mean of the worst (1-a) *100% drawdowns. The CDaR risk function contains the maximal drawdown and average drawdown as its limiting cases. For a particular example, we find optimal portfolios with constraints on the maximal drawdown, average drawdown and several intermediate cases between these two. The CDaR family of risk functions originates from the conditional value-at-risk (CVaR) measure. Some recommendations on how to select the optimal risk measure for getting practically stable portfolios are provided. We solved a real life portfolio allocation problem using the proposed risk functions. 1. Introduction Optimal portfolio allocation is a longstanding issue in both practical portfolio management and academic research on portfolio theory. Various methods have been proposed and studied (for a review, see, for example, Grinold and Kahn, 1999). All of them, as a starting point, assume some measure of portfolio risk. From a standpoint of a fund manager, who trades clients' or bank’s proprietary capital, and for whom the clients' accounts are the only source of income coming in the form of management and incentive fees, losing these accounts is equivalent to the death of his business. -
Asset and Risk Allocation Policy
____________________________________________________________________________ UNIVERSITY OF CALIFORNIA GENERAL ENDOWMENT POOL ASSET AND RISK ALLOCATION POLICY Approved March 15, 2018 ______________________________________________________________________________ UNIVERSITY OF CALIFORNIA GENERAL ENDOWMENT POOL ASSET AND RISK ALLOCATION POLICY POLICY SUMMARY/BACKGROUND The purpose of this Asset and Risk Allocation Policy (“Policy”) is to define the asset types, strategic asset allocation, risk management, benchmarks, and rebalancing for the University of California General Endowment Pool (“GEP”). The Investments Subcommittee has consent responsibilities over this policy. POLICY TEXT ASSET CLASS TYPES Below is a list of asset class types in which the GEP may invest so long as they do not conflict with the constraints and restrictions described in the GEP Investment Policy Statement. The criteria used to determine which asset classes may be included are: Positive contribution to the investment objective of GEP Widely recognized and accepted among institutional investors Low cross correlations with some or all of the other accepted asset classes Based on the criteria above, the types of assets for building the portfolio allocation are: 1. Public Equity Includes publicly traded common and preferred stock of issuers domiciled in US, Non-US, and Emerging (and Frontier) Markets. The objective of the public equity portfolio is to generate investment returns with adequate liquidity through a globally diversified portfolio of common and preferred stocks. 2. Liquidity (Income) Liquidity includes a variety of income related asset types. The portfolio will invest in interest bearing and income based instruments such as corporate and government bonds, high yield debt, emerging markets debt, inflation linked securities, cash and cash equivalents. The portfolio can hold a mix of traditional (benchmark relative) strategies and unconstrained (benchmark agnostic) strategies. -
Tracking Error
India Index Services & Products Ltd. Tracking Error TRACKING ERROR An Index Fund is a mutual fund scheme that invests in the securities in the target Index in the same proportion or weightage of the securities as it bears to the target index. The investment objective of an index fund is to achieve returns which are commensurate to that of the target Index. An investment manager attempts to replicate the investment results of the target index by holding all the securities in the Index. Though Index Funds are designed to provide returns that closely track the benchmarked Index, Index Funds carry all the risks associated with the type of asset the fund holds. Indexing merely ensures that the returns of the Index Fund will not stray far from the returns on the Index that the fund mimics. Still there are instances, which are mentioned below, that leads to mismatch of the returns of the index with that of the fund. This mismatch or the difference in the returns of the Index with that of the fund is known as Tracking Error. Tracking Error Tracking error is defined as the annualised standard deviation of the difference in returns between the Index fund and its target Index. In simple terms, it is the difference between returns from the Index fund to that of the Index. An Index fund manager needs to calculate his tracking error on a daily basis especially if it is open-ended fund. Lower the tracking error, closer are the returns of the fund to that of the target Index. -
T013 Asset Allocation Minimizing Short-Term Drawdown Risk And
Asset Allocation: Minimizing Short-Term Drawdown Risk and Long-Term Shortfall Risk An institution’s policy asset allocation is designed to provide the optimal balance between minimizing the endowment portfolio’s short-term drawdown risk and minimizing its long-term shortfall risk. Asset Allocation: Minimizing Short-Term Drawdown Risk and Long-Term Shortfall Risk In this paper, we propose an asset allocation framework for long-term institutional investors, such as endowments, that are seeking to satisfy annual spending needs while also maintaining and growing intergenerational equity. Our analysis focuses on how asset allocation contributes to two key risks faced by endowments: short-term drawdown risk and long-term shortfall risk. Before approving a long-term asset allocation, it is important for an endowment’s board of trustees and/or investment committee to consider the institution’s ability and willingness to bear two risks: short-term drawdown risk and long-term shortfall risk. We consider various asset allocations along a spectrum from a bond-heavy portfolio (typically considered “conservative”) to an equity-heavy portfolio (typically considered “risky”) and conclude that a typical policy portfolio of 60% equities, 30% bonds, and 10% real assets represents a good balance between these short-term and long-term risks. We also evaluate the impact of alpha generation and the choice of spending policy on an endowment’s expected risk and return characteristics. We demonstrate that adding alpha through manager selection, tactical asset allocation, and the illiquidity premium significantly reduces long-term shortfall risk while keeping short-term drawdown risk at a desired level. We also show that even a small change in an endowment’s spending rate can significantly impact the risk profile of the portfolio. -
Performance Measurement in Finance : Firms, Funds and Managers
PERFORMANCE MEASUREMENT IN FINANCE Butterworth-Heinemann Finance aims and objectives • books based on the work of financial market practitioners and academics • presenting cutting edge research to the professional/practitioner market • combining intellectual rigour and practical application • covering the interaction between mathematical theory and financial practice • to improve portfolio performance, risk management and trading book performance • covering quantitative techniques market Brokers/Traders; Actuaries; Consultants; Asset Managers; Fund Managers; Regulators; Central Bankers; Treasury Officials; Technical Analysts; and Academics for Masters in Finance and MBA market. series titles Return Distributions in Finance Derivative Instruments: theory, valuation, analysis Managing Downside Risk in Financial Markets: theory, practice and implementation Economics for Financial Markets Global Tactical Asset Allocation: theory and practice Performance Measurement in Finance: firms, funds and managers Real R&D Options series editor Dr Stephen Satchell Dr Satchell is Reader in Financial Econometrics at Trinity College, Cambridge; Visiting Professor at Birkbeck College, City University Business School and University of Technology, Sydney. He also works in a consultative capacity to many firms, and edits the journal Derivatives: use, trading and regulations. PERFORMANCE MEASUREMENT IN FINANCE Firms, Funds and Managers Edited by John Knight Stephen Satchell OXFORD AMSTERDAM BOSTON LONDON NEW YORK PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO -
A Nonlinear Optimisation Model for Constructing Minimal Drawdown Portfolios
A nonlinear optimisation model for constructing minimal drawdown portfolios C.A. Valle1 and J.E. Beasley2 1Departamento de Ci^enciada Computa¸c~ao, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-010, Brazil [email protected] 2Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK [email protected] August 2019 Abstract In this paper we consider the problem of minimising drawdown in a portfolio of financial assets. Here drawdown represents the relative opportunity cost of the single best missed trading opportunity over a specified time period. We formulate the problem (minimising average drawdown, maximum drawdown, or a weighted combination of the two) as a non- linear program and show how it can be partially linearised by replacing one of the nonlinear constraints by equivalent linear constraints. Computational results are presented (generated using the nonlinear solver SCIP) for three test instances drawn from the EURO STOXX 50, the FTSE 100 and the S&P 500 with daily price data over the period 2010-2016. We present results for long-only drawdown portfolios as well as results for portfolios with both long and short positions. These indicate that (on average) our minimal drawdown portfolios dominate the market indices in terms of return, Sharpe ratio, maximum drawdown and average drawdown over the (approximately 1800 trading day) out-of-sample period. Keywords: Index out-performance; Nonlinear optimisation; Portfolio construction; Portfo- lio drawdown; Portfolio optimisation 1 Introduction Given a portfolio of financial assets then, as time passes, the price of each asset changes and so by extension the value of the portfolio changes. -
Maximum Drawdown Is Generally Greater Than in IID Normal Case
THE HIGH COST OF SIMPLIFIED MATH: Overcoming the “IID Normal” Assumption in Performance Evaluation Marcos López de Prado Hess Energy Trading Company Lawrence Berkeley National Laboratory Electronic copy available at: http://ssrn.com/abstract=2254668 Key Points • Investment management firms routinely hire and fire employees based on the performance of their portfolios. • Such performance is evaluated through popular metrics that assume IID Normal returns, like Sharpe ratio, Sortino ratio, Treynor ratio, Information ratio, etc. • Investment returns are far from IID Normal. • If we accept first-order serial correlation: – Maximum Drawdown is generally greater than in IID Normal case. – Time Under Water is generally longer than in IID Normal case. – However, Penance is typically shorter than 3x (IID Normal case). • Conclusion: Firms evaluating performance through Sharpe ratio are firing larger numbers of skillful managers than originally targeted, at a substantial cost to investors. 2 Electronic copy available at: http://ssrn.com/abstract=2254668 SECTION I The Need for Performance Evaluation Electronic copy available at: http://ssrn.com/abstract=2254668 Why Performance Evaluation? • Hedge funds operate as banks lending money to Portfolio Managers (PMs): – This “bank” charges ~80% − 90% on the PM’s return (not the capital allocated). – Thus, it requires each PM to outperform the risk free rate with a sufficient confidence level: Sharpe ratio. – This “bank” pulls out the line of credit to underperforming PMs. Allocation 1 … Investors’ funds Allocation N 4 How much is Performance Evaluation worth? • A successful hedge fund serves its investors by: – building and retaining a diversified portfolio of truly skillful PMs, taking co-dependencies into account, allocating capital efficiently. -
TRACKING ERROR: an Essential Tool in Evaluating Index Funds
JOURNAL OF INVESTMENT CONSULTING TRACKING ERROR: An Essential Tool in Evaluating Index Funds By Chris Tobe, CFA As the dollars being indexed grow and portfolio management becon1es more sophisticated, consultants should demand tighter tracking to the index in question from their index n1anagers. In fact, the key to assessiI1g index fund managers is determining if their performance has tracked their benchmarks within reasonable tolerances. The alltl10r reviews what is known about this aspect of performance evaluation. He also sl1ares 11is experie11ce in detern1ining what factors caused a difference in performance for a $3 billion S&P 500 index fund. This article is based on a study written by the author and should be no attempt to outperform the index or Dr. Ken Miller for Kentucky State Auditor Edward B. bencl1mark. William F Sharpe holds that a passive Hatchett Jr. The study was reported in the July 27, 1998, strategy requires the manager issue of Pensions and Investments. Earlier versions of this ... to hold every security from the market, with each article were presented at two conferences on indexing. represented in the same manner as in the n1arket. Thus, if security X represents 3 percent of the value ver $1 trillion is now being managed of the securities in the market, a passive investor's within index funds. Lipper lists the portfolio will have 3 percent of its value invested in Vanguard 500 Index fund as the second X. Equivalently; a passive manager will hold the same percentage of the total outstanding amount of largest stock fllnd in the world with a each security in the market. -
Capital Adequacy Requirements (CAR)
Guideline Subject: Capital Adequacy Requirements (CAR) Chapter 3 – Credit Risk – Standardized Approach Effective Date: November 2017 / January 20181 The Capital Adequacy Requirements (CAR) for banks (including federal credit unions), bank holding companies, federally regulated trust companies, federally regulated loan companies and cooperative retail associations are set out in nine chapters, each of which has been issued as a separate document. This document, Chapter 3 – Credit Risk – Standardized Approach, should be read in conjunction with the other CAR chapters which include: Chapter 1 Overview Chapter 2 Definition of Capital Chapter 3 Credit Risk – Standardized Approach Chapter 4 Settlement and Counterparty Risk Chapter 5 Credit Risk Mitigation Chapter 6 Credit Risk- Internal Ratings Based Approach Chapter 7 Structured Credit Products Chapter 8 Operational Risk Chapter 9 Market Risk 1 For institutions with a fiscal year ending October 31 or December 31, respectively Banks/BHC/T&L/CRA Credit Risk-Standardized Approach November 2017 Chapter 3 - Page 1 Table of Contents 3.1. Risk Weight Categories ............................................................................................. 4 3.1.1. Claims on sovereigns ............................................................................... 4 3.1.2. Claims on unrated sovereigns ................................................................. 5 3.1.3. Claims on non-central government public sector entities (PSEs) ........... 5 3.1.4. Claims on multilateral development banks (MDBs) -
Hedging Climate Risk Mats Andersson, Patrick Bolton, and Frédéric Samama
AHEAD OF PRINT Financial Analysts Journal Volume 72 · Number 3 ©2016 CFA Institute PERSPECTIVES Hedging Climate Risk Mats Andersson, Patrick Bolton, and Frédéric Samama We present a simple dynamic investment strategy that allows long-term passive investors to hedge climate risk without sacrificing financial returns. We illustrate how the tracking error can be virtually eliminated even for a low-carbon index with 50% less carbon footprint than its benchmark. By investing in such a decarbonized index, investors in effect are holding a “free option on carbon.” As long as climate change mitigation actions are pending, the low-carbon index obtains the same return as the benchmark index; but once carbon dioxide emissions are priced, or expected to be priced, the low-carbon index should start to outperform the benchmark. hether or not one agrees with the scientific the climate change debate is not yet fully settled and consensus on climate change, both climate that climate change mitigation may not require urgent Wrisk and climate change mitigation policy attention. The third consideration is that although the risk are worth hedging. The evidence on rising global scientific evidence on the link between carbon dioxide average temperatures has been the subject of recent (CO2) emissions and the greenhouse effect is over- debates, especially in light of the apparent slowdown whelming, there is considerable uncertainty regarding in global warming over 1998–2014.1 The perceived the rate of increase in average temperatures over the slowdown has confirmed the beliefs of climate change next 20 or 30 years and the effects on climate change. doubters and fueled a debate on climate science There is also considerable uncertainty regarding the widely covered by the media.