Introduction to Risk Parity and Budgeting

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Although portfolio management didn't change much during the 40 years after the seminal works of Markowitz and Sharpe, the development of risk budgeting techniques marked an important milestone in the deepening of the relationship between risk and asset management. Risk parity then became a popular f
Chapman hall/ CRC FINANCIAL MATHEMATICS SERIES Introduction to Risk Parity and Budgeting Thierry roncalli CRC) CRC Press Boca raton London New york CRC Press is an imprint of the Taylor Francis Group, an informa business a chapman hall book K21545 FMindd 5 6/7/132:49PM CRC Press Taylor Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca raton fl 334 87-2742 o 2014 by Taylor Francis Group, LLC CRC Press is an imprint of Taylor Francis Group, an Informa business No claim to original U.S. Government works Version date: 20130607 International Standard Book Number-13: 978-1-4822-0716-3(eBook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor francis Web site at and the crc press Web site at chapman hall/crc Financial mathematics series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector This series aims to capture new developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks reference works and handbooks that are meant to appeal to both academics and practitioners The inclusion of numerical code and concrete real world examples is highly encouraged Series editors M.A. H Dempster Dilip b. madan Rama Cont Centre for Financial Research Robert H. Smith school Department of Mathematics Department of pure of Business Imperial college Mathematics and statistics University of maryland University of cambridge Published titles American-Style Derivatives; Valuation and computation, Jerome detemple Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing, Pierre henry-Labordere Computational Methods in Finance, Ali hirsa Credit risk: Models, Derivatives, and Management, Niklas Wagner Engineering BGM, Alan brace Financial Modelling with Jump Processes, Rama Cont and Peter Tankov Interest Rate Modeling: Theory and practice, Lixin Wu Introduction to Credit Risk modeling, Second Edition, Christian bluhm, Ludger Overbeck, and Christoph Wagner An Introduction to Exotic Option Pricing, Peter Buchen Introduction to Risk Parity and budgeting Thierry roncalli Introduction to Stochastic Calculus applied to inance second edition Damien lamberton and Bernard lapeyre Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn, and gerald kroisandt Monte Carlo Simulation with Applications to Finance, Hui Wang Nonlinear Option Pricing, Julien guyon and Pierre Henry-Labordere Numerical Methods for Finance, John A D. Appleby, David C. edelman and John,h. miller Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn Portfolio optimization and performance analysis, Jean-Luc Prigent Quantitative Fund Management, M.A. H. Dempster, Georg Pug, and Gautam mitra K21545 FM indd 6/7/132:49PM Risk analysis in Finance and Insurance, Second Edition, Alexander melnikov Robust libor modelling and pricing of derivative products John schoenmakers Stochastic Finance: A Numeraire Approach, Jan veer Stochastic Financial Models, Douglas Kennedy Stochastic Processes with Applications to Finance, Second Edition Masaaki kijima Structured Credit Portfolio analysis, Baskets cdos, Christian bluhm and Ludger Overbeck Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy Unravelling the credit Crunch, David Murphy Proposals for the series should be submitted to one of the series editors above or directly to CRC Press, Taylor francis group 3 Park Square, Milton Park abingdon Oxfordshire oX14 4RN UK K21545 FM indd 3 6/7/132:49PM K21545 FMindd 4 6/7/132:49PM Introduction The death of markowitz optimization? For a long time, investment, theory and practice has been summarized as follows. The capital asset pricing model stated that the market portfolio is op timal. during the 1990s, the development of passive management confirmed the work done by william Sharpe. At that same time. the number of insti tutional investors grew at an impressive pace. many of these investors used pa assivc man agement for thcir cquity and bond exposures For assct allocation they used the optimization model developed by Harry Markowitz, even though t hey knew that such an approach was very sensitive to input parameters, and n particular, to expected returns(Merton, 1980). One reason is that there was no other alternative model. Another reason is that the markowitz model is easy to use and simple to explain. For expected returns, these investors generally considered long-term historical figures, stating that past history can serve as a reliable guide for the future. Management boards of pension funds were won over by this scientific approach to asset allocation The first serious warning shot came with the dot-com crisis. Some insti- tutional investors, in particular defined benefit pension plans, lost substan tial amounts of money because of their high exposure to equities(Ryan and Fabozzi, 2002). In N ovembe 2001, the pension plan of The Boots Company, a UK pharmacy retailer, decided to invest 100% in bonds(Sutcliffe, 2005) Nevertheless, the performance of the equity market between 2003 and 2007 restored confidence that standard financial models would continue to work and that the dot-com crisis was a non-recurring exception. Ilowever, the 2008 fina ncia.l crisis highlighted the risk in herent. in many strategic asset a locations Moreover, for institutional investors, the crisis was unprecedentedly severe. In 2000, the internet crisis was limited to large capitalization stocks and certain sectors. Small capitalizations and value stocks were not affected while the performance of hedge funds was flat. In 2008, the subprime crisis led to a violent drop in credit strategies and asset-backed securities. Equities posted negative returns of about -50%. The performance of hedge funds and alterna tive assets was poor. There was also a paradox. Many institutional investors diversified their portfolios by considering several asset classes and different re gions. Unfortunately, this diversification was not enough to protect them. In the end, the 2008 financial crisis was Inore dainlaginlg than the dot-COlll crisis This was particularly true for institutional investors in continental europe who were relatively well protected against the collapse of the internet bubble because of their low exposure to equities. This is why the 2008 financial crisis was a deep trauma for world-wide institutional investors Most institutional portfolios wcrc calibrated through portfolio optimiza tion. In this context, Markowitz's modern portfolio theory was strongly crit icized by professionals, and several journal articles announced the death of the Markowitz model. These extreme reactions can be explained by the fact that diversification is traditionally associated with Markowitz optimization and it failed during the financial crisis. However, the problem was not entirely due to the allocation method. Indeed, much of the failure was caused by the input parameters With expected returns calibrated to past figures, the model induced an overweight in equities. It also promoted assets that were supposed to have a low correlation to equities. Nonetheless, correlations between as set classes increased significantly during the crisis. In the end, the promised diversification did not occur Today, it is hard to find investors who defend Markowitz optimization However, the criticisms concern not so much the model itself but the way it is used. In the 1990s, researchers began to develop regularization techniques to limit the impact of estimation errors in input parameters and many im provcmcnts havc bccn madc in rcccnt ycars. In addition, wc now havc a bettor understanding of how this model works. Moreover. we also have a theoreti cal framework to measure the impact of constraints (Jagannathan and Ma 2003 ). More recently, robust optimization based on the lasso approach has im- proved optimized portfolios(DeMiguel et al., 2009). So the Markowitz model is certainly not dead. Investors must understand that it is a fabulous tool for combining risks and expected returns. The goal of Markowitz optimization is to find arbitrage factors and build a portfolio that will play on them. By constructioN, this approach is all aggressive Inodel of active Inlallagenent. IIl this case, it is normal that the model should be sensitive to input param eters(Green and Hollifield, 1992. Changing the parameter values modifies the implied bets. Accordingly, if input parameters are wrong, then arbitrage factors and bets are also wrong, and the resulting portfolio is not satisfied If investors want a more defensive model, they have to define less aggressive parameter values. This is the main message behind portfolio regularization In consequence. reports of the death of the Markowitz model have been greatly exaggerated, because it will continue to be used intensively in active manage mcnt strategics Morcovcr, thcrc arc no othcr serious and powerful modcls to take into account return forecasts 1 See for example the article "Is Markowitz Dead? Goldman Thinks So"published in Decenber 2012 by AsiallIrvestor The rise of risk parity portfolios There are different ways to obtain less aggressive active portfolios. The first one is to use less aggressive paraineters. For instance, if we assullle that expected returns are the same for all of the assets we obtain the minimum variance(or MV) portfolio. The second way is to use heuristic methods of asset allocation. The term heuristic'refers to experience-based techniques and trial and-error methods to find an acceptable solution, which does not correspond to the optimal solution of an optimization problem. The equally weighted (o EW)portfolio is an example of such non-optimized rule of thumb' portfolio By allocating the saine weight to all the assets of the investiment universe we considerably reduce the sensitivity to input parameters. In fact, there are no activc bcts any longcr. Although these two allocation mcthods have bccn known for a long time, they only became popular after the collapse of the internet bubble Risk parity is another example of heuristic methods. The underlying idea is to build a halanced port folio in such a way that the risk contribution is t, he same for different assets. It is then an equally weighted portfolio in terms of isk, not in terms of weights. Like the minimum variance and equally weighted portfolios, it is impossible to date the risk parity portfolio. The term risk parity was coined by Qian(2005 ) However, the risk parity approach was certainly used before 2005 by some CTA and equity market neutral funds. For instance it was the core approach of the All Weather fund managed by Bridgewater for many years(Da.lio, 2004). At this point we note that the risk parity port folio is used, because it makes sense from a practical point of view. However, it was not until the theoretical work of Maillard et al.(2010), first published in 2008 that the analytical properties were explored. In particular, they showed that this portfolio exists, is unique and is located between the minimum variance and equally weighted portfolios Since 2008, we have observed an increasing popularity of the risk parity portfolio. For example, Journal of Investing and Investment and Pensions Eu- rope(Ipe) ran special issues on risk parity in 2012. In the same year, The Financial Times and Wall Street Journal published several articles on this topic In fact today, the term risk parity covers diffcrcnt allocation mcth ods. For instance, some professionals use the term risk parity when the asset weight is inversely proportional to the asset return volatility. Others consider that the risk parity portfolio corresponds to the equally weighted risk con- tribution( or eRC)portfolio. Sometimes, risk parity is equivalent to a risk budgeting (or RB)portfolio. In this case, the risk budgets are not necessarily the same for all of the assets that compose the portfolio. Initially, risk parity 2New Allocation Funds Redefine Idea of Balance"(February 2012),"Same Returns Less Risk"(June 2012), "Risk Parity Strategy Has Its Critics as Well as Fans"(June 2012) "Investors Rush for Risk Parity Shield"(Septerber 2012),etc

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