An EDHEC-Risk Institute Publication          New Frontiers in        Benchmarking andLiability-Driven Investing           ...
This publication has benefitted from research conducted as part of numerous EDHEC-Risk Institute research programmes and  ...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010Table of ContentsAbstract ......................
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                          Abo...
AbstractAn EDHEC-Risk Institute Publication   5
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                          Abs...
2. xxxxxxxxxxxxxxxxxx            Introduction              An EDHEC-Risk Institute Publication   7
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           In...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010Introductionown challenges and difficulties, ...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           In...
1. Asset Allocation and Portfolio        Construction Decisions in       the Optimal Design of the  Performance-Seeking Po...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           1....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           1....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           1....
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                  1. Asset Al...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           1....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           1....
2. Asset Allocation and Portfolio   Construction Decisions in the           Optimal Design of the      Liability-Hedging P...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           2....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           2....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           2....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio Constructio...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           2....
3. Dynamic Allocation Decisionsto the Performance-Seeking and     Liability-Hedging Portfolios                      An EDH...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                           3....
New Frontiers in Benchmarking and Liability-Driven Investing - September 20103. Dynamic Allocation Decisions to thePerform...
New Frontiers in Benchmarking and Liability-Driven Investing - September 2010                                            3...
New Frontiers in Benchmarking and Liability-Driven Investing - September 20103. Dynamic Allocation Decisions to thePerform...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, E...
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New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, EDHEC Business School

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Professor Noël Amenc, Director, EDHEC Risk Institute, Professor of Finance at the EDHEC Business School gave his presentation titled "New Frontiers in Benchmarking and Liability-Driven Investing" at the Middle East Investments Summit.

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New Frontiers in Benchmarking and Liability-Driven Investing - Presentation: Noël Amenc, Director, EDHEC Risk Institute, EDHEC Business School

  1. 1. An EDHEC-Risk Institute Publication New Frontiers in Benchmarking andLiability-Driven Investing September 2010 Institute
  2. 2. This publication has benefitted from research conducted as part of numerous EDHEC-Risk Institute research programmes and chairs, notably the "Asset-Liability Management and Institutional Investment Management" research chair in partnership with BNP Paribas Investment Partners and the "Dynamic Allocation Models and New Forms of Target-Date Funds" research chair in partnership with UFG-LFP. Printed in France, September 2010. Copyright© EDHEC 2010.2 The opinions expressed in this study are those of the authors and do not necessarily reflect those of EDHEC Business School. The authors can be contacted at research@edhec-risk.com.
  3. 3. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010Table of ContentsAbstract ................................................................................................................... 5Introduction ........................................................................................................... 71. Asset Allocation and Portfolio Construction Decisionsin the Optimal Design of the Performance-Seeking Portfolio ....................... 112. Asset Allocation and Portfolio Construction Decisionsin the Optimal Design of the Liability-Hedging Portfolio ............................. 213. Dynamic Allocation Decisions to the Performance-Seekingand Liability-Hedging Portfolios ....................................................................... 29Conclusion .............................................................................................................. 43Appendix ................................................................................................................. 45References .............................................................................................................. 51About EDHEC-Risk Institute ............................................................................... 57EDHEC-Risk Institute Publications and Position Papers (2007-2010) ......... 61 An EDHEC-Risk Institute Publication 3
  4. 4. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 About the Authors Noël Amenc is professor of finance and director of EDHEC-Risk Institute. He has a masters in economics and a PhD in finance and has conducted active research in the fields of quantitative equity management, portfolio performance analysis, and active asset allocation, resulting in numerous academic and practitioner articles and books. He is a member of the editorial board of the Journal of Portfolio Management, associate editor of the Journal of Alternative Investments and a member of the scientific advisory council of the AMF (French financial regulatory authority). Lionel Martellini is professor of finance at EDHEC Business School and scientific director of EDHEC-Risk Institute. He has graduate degrees in economics, statistics, and mathematics, as well as a PhD in finance from the University of California at Berkeley. Lionel is a member of the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. An expert in quantitative asset management and derivatives valuation, Lionel has published widely in academic and practitioner journals and has co-authored textbooks on alternative investment strategies and fixed-income securities. Felix Goltz is head of applied research at EDHEC-Risk Institute. He does research in empirical finance and asset allocation, with a focus on alternative investments and indexing strategies. His work has appeared in various international academic and practitioner journals and handbooks. He obtained a PhD in finance from the University of Nice Sophia-Antipolis after studying economics and business administration at the University of Bayreuth and EDHEC Business School. Vincent Milhau holds masters degrees in statistics (ENSAE) and financial mathematics (Université Paris VII), as well as a PhD in finance (Université de Nice-Sophia Antipolis). In 2006, he joined EDHEC-Risk Institute, where he is currently a senior research engineer. His research focus is on portfolio selection problems and continuous-time asset-pricing models.4 An EDHEC-Risk Institute Publication
  5. 5. AbstractAn EDHEC-Risk Institute Publication 5
  6. 6. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 Abstract Meeting the challenges of modern investment practice involves the design of novel forms of investment solutions, as opposed to investment products, customised to meet investors long-term objectives while respecting the short-term (regulatory or otherwise) constraints they have to face. We argue in this paper that such new forms of investment solutions should rely on the use of improved performance-seeking and liability-hedging building-block portfolios, as well as on the use of improved dynamic allocation strategies. Although each of the ingredients discussed in this paper may already be found separately in existing investment products, we suggest that it is only by putting the pieces of the puzzle together, and by combining the underlying sources of expertise and added value that the asset management industry will satisfactorily address investors needs.6 An EDHEC-Risk Institute Publication
  7. 7. 2. xxxxxxxxxxxxxxxxxx Introduction An EDHEC-Risk Institute Publication 7
  8. 8. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 Introduction Asset management is justified as an industry approaches to optimal spending of investors’ by the capacity of adding value through the risk budgets. design of investment solutions that match investors needs. For more than fifty years, Academic research has provided very the industry has in fact focused mostly useful guidance to the ways asset on security selection as a single source allocation and portfolio construction of added value. This focus has somewhat decisions should be analysed so as to best distracted the industry from another key improve investors’ welfare. In brief, the source of added value, namely, portfolio “fund separation theorems” that lie at the construction and asset allocation decisions. core of modern portfolio theory advocate In the face of recent crises, and given the separate management of performance intrinsic difficulty of delivering added value and risk-control objectives. In the through security selection decisions alone, context of asset allocation decisions with the relevance of the old paradigm has been consumption/liability objectives, it can be questioned with heightened intensity, and shown that the suitable expression of the a new paradigm is starting to emerge. fund separation theorem provides rational support for liability-driven investment1 - More generally, thereare other forms of hedging In a nutshell, the new paradigm recognises (LDI) techniques that have recently beendemand that allow investors that the art and science of portfolio promoted by a number of investmentto neutralise the impactof unexpected changes in management consists of constructing banks and asset management firms. Theserisk factors affecting theopportunity set or the wealth dedicated portfolio solutions, as opposed solutions involve, on the one hand, theprocess. This is discussed in to one-size-fits-all investment products, so design of a customised liability-hedgingmore detail later in this paper,where we cover life-cycle as to reach the return objectives defined portfolio (LHP), the sole purpose of whichinvestment strategies. by the investor, while respecting the is to hedge away as effectively as possible investors constraints expressed in terms the impact of unexpected changes in risk of (absolute or relative) risk budgets. In factors affecting liability values (most this broader context, asset allocation and notably interest rate and inflation risks), portfolio construction decisions appear and, on the other hand, the design of a as the main source of added value by performance-seeking portfolio (PSP), whose the investment industry, with security raison d’être is to provide investors an selection being a third-order problem. optimal risk/return trade-off.1 As argued throughout this paper, asset allocation and portfolio construction One of the implications of this LDI decisions are intimately related to risk paradigm is that one should distinguish management. In the end, the quintessence two different levels of asset allocation of investment management is essentially decisions: allocation decisions involved in about finding optimal ways to spend risk the design of the performance-seeking or budgets that investors are reluctantly the liability-hedging portfolio (design of willing to set, with a focus on allowing better building blocks, or BBBs), and asset the greatest possible access to performance allocation decisions involved in the optimal potential while respecting such risk budgets. split between the PSP and the LHP (design Risk diversification, risk hedging, and risk of advanced asset allocation decisions, or insurance will be shown to be three useful AAAs). Each level of analysis involves its8 An EDHEC-Risk Institute Publication
  9. 9. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010Introductionown challenges and difficulties, and whilethe LDI paradigm is now widely adoptedin the institutional world, very few marketparticipants adopt an implementationapproach of the paradigm that is fullyconsistent with the state-of-the-art ofacademic research.Our ambition in this paper is to describe themost advanced forms of LDI strategies. Ourfocus is to provide not so much a thoroughand rigorous treatment of all technicalquestions related to asset allocation andportfolio construction as a holistic overviewof the key conceptual challenges involved.We address both questions (BBB and AAA)in this paper. More specifically, we firstfocus here on how to construct efficientperformance-seeking and liability-hedgingportfolios, and then move on to provideinformation on how to allocate optimallyto these two building blocks once they havebeen designed.In the next section, we present thechallenges related to asset allocation andportfolio construction decisions withinthe PSP. We then discuss the challengesrelated to asset allocation and portfolioconstruction decisions within the LHP. Thelast section provides an introduction tooptimal allocation to the PSP and the LHPfor a long-term investor facing short-termconstraints, once these two key buildingblocks have been properly designed. A fewconcluding thoughts can be found in afinal section. Further technical details arerelegated to an appendix. An EDHEC-Risk Institute Publication 9
  10. 10. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 Introduction10 An EDHEC-Risk Institute Publication
  11. 11. 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio An EDHEC-Risk Institute Publication 11
  12. 12. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio Modern portfolio theory again provides In this context, it is straightforward to some useful guidance to the optimal design show by standard arguments that the only of a PSP that would best suit investors’ efficient portfolio composed with risky needs. More precisely, the prescription is assets is the maximum Sharpe ratio portfolio, that the PSP should be obtained as the also known as the tangency portfolio. result of a portfolio optimisation procedure Appendix A.1 provides more details. aiming at generating the highest risk-reward ratio. Finally, the Sharpe ratio reads (where we further let e be a vector of ones of size Portfolio optimisation is a straightforward N): procedure, at least in principle. In a mean-variance setting, for example, the prescription consists of generating a maximum Sharpe ratio (MSR) portfolio And the optimal portfolio is given by: based on expected return, volatility, and pairwise correlation parameters for all assets to be included in the portfolio, a procedure which can even be handled analytically in the absence of portfolio constraints. (1) More precisely, consider a simple mean-variance problem: This is a two-fund separation theorem, 1 2 which gives the allocation to the MSR max μp − γσ p performance-seeking portfolio (PSP), with w 2 the rest invested in cash, as well as the Here, the control variable is a vector w composition of the MSR performance- of optimal weight allocated to various seeking portfolio. risky assets, µp is the portfolio expected return, and σp the portfolio volatility. We In practice, investors end up holding more further assume that the investor is facing or less imperfect proxies for the truly the following investment opportunity set: optimal performance-seeking portfolio, if a riskless bond paying the risk-free rate r, only because of the presence of parameter and a set of N risky assets with expected uncertainty, which makes it impossible to return vector µ (of size N) and covariance obtain a perfect estimate for the maximum matrix Σ (of size NxN), all assumed constant Sharpe ratio portfolio. If we let λ be the so far. Sharpe ratio of the (generally inefficient) PSP actually held by the investor, and σ With these notations, the portfolio expected be its volatility, we obtain the following return and volatility respectively are given optimal allocation strategy: by: (2)12 An EDHEC-Risk Institute Publication
  13. 13. New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of thePerformance-Seeking PortfolioHence, the allocation to the performance- which is typically delegated to professionalseeking portfolio is a function of two money managers, the portfolio constructionobjective parameters, the PSP volatility step. On the other hand, when the MSRand the PSP Sharpe ratio, and one subjective proxies are obtained for each asset class,parameter, the investor’s risk aversion. The an optimal allocation to the various assetoptimal allocation to the PSP is inversely classes is eventually generated so as toproportional to the investor’s risk-aversion. generate the maximum Sharpe ratio at theIf risk aversion rises to infinity, the investor global portfolio level. This step is calledholds only the risk-free asset, as should the asset allocation step, and it is typicallybe expected. For finite risk aversion, the handled by a centralised decision makerallocation to the PSP is inversely proportional (e.g., a pension fund CIO) with or withoutto the PSP volatility, and it is proportional the help of specialised consultants, asto the PSP Sharpe ratio. As a result, if the opposed to being delegated to decentralisedSharpe ratio of the PSP is increased, one asset managers. We discuss both of thesecan invest more in risky assets. Hence, steps in what follows.risk management is not only about riskreduction; it is also about performanceenhancement through a better expenditure 1.1. Portfolio Construction Step:of investors’ risk budgets. We revisit this Designing Efficient Benchmarkspoint later in the paper. In the absence of active views, the default option consists of using market-cap-The expression (1) is useful because, in weighted indices as proxies for the assetprinciple, it provides a straightforward class MSR portfolio. Academic research,expression for the optimal portfolio however, has found that such indices werestarting from a set of N risky assets. In the likely to be severely inefficient portfoliospresence of a realistically large number N (Haugen and Baker 1991; Grinold 1992;of securities, the curse of dimensionality, Amenc, Goltz, and Le Sourd 2006). In sum,however, makes it practically impossible for market-cap-weighted indices are not goodinvestors to implement such direct one-step choices as investment benchmarks becauseportfolio optimisation decisions involving they are poorly diversified portfolios. Inall individual components of the asset fact, capitalisation weighting tends tomix. The standard alternative approach lead to exceedingly high concentration inwidely adopted in investment practice relatively few stocks. As a result of theirconsists instead of first grouping individual lack of diversification, cap-weighted indicessecurities in various asset classes by various have empirically been found to be highlydimensions, e.g., country, sector, and/or inefficient portfolios that do not providestyle within the equity universe, or country, investors fair rewards for the risks theymaturity, and credit rating within the bond take. For the same reason, they have beenuniverse, and subsequently generating the found to be dominated by equally-weightedoptimal portfolio through a two-stage benchmarks (De Miguel, Garlappi, and Uppalprocess. On the one hand, investable proxies 2009); equally-weighted benchmarks useare generated for maximum Sharpe ratio a naive weighting scheme that ensures(MSR) portfolios within each asset class in that they are well diversified, but theythe investment universe. We call this step, are optimal in the mean-variance sense An EDHEC-Risk Institute Publication 13
  14. 14. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio if and only if all securities have identical expected return estimates should include expected returns and volatilities and all systematic risk measures, idiosyncratic risk pairs of correlation are identical. measures, and downside risk measures. In what follows, we analyse in some detail 1.1.1. Robust Estimators for a number of alternatives based on practical Covariance Parameters implementation of modern portfolio theory In practice, successful implementation of that have been suggested to generate more a theoretical model relies not only on its efficient proxies for the MSR portfolio in conceptual grounds but also on the reliability the equity or fixed-income investment of the input to the model. In mean-variance universes (see exhibit 1). (MV) optimisation the results will depend greatly on the quality of the parameter Modern portfolio theory was born with the estimates: the covariance matrix and the efficient frontier analysis of Markowitz in expected returns of assets. 1952. Unfortunately, early applications of the technique, based on naïve estimates of Several improved estimates for the the input parameters, have been of little use covariance matrix have been proposed, 2 - Another key challenge is the presence of because they lead to unreasonable portfolio including most notably the factor-based non-stationary risk allocations. approach (Sharpe 1963), the constant parameters, which can be accounted for with correlation approach (Elton and Gruber conditional factor models capturing time-dependencies We explain below first how to help bridge 1973), and the statistical shrinkage (e.g., GARCH-type models) the gap between portfolio theory and approach (Ledoit and Wolf 2004). and state-dependencies (e.g., Markov regime-switching portfolio construction by showing how to In addition, Jagannathan and Ma (2003) models) in risk parameter estimates. generate enhanced parameter estimates find that imposing (non-short selling) and to improve the quality of the portfolio constraints on the weights in the optimisation outputs (optimal portfolio optimisation programme improves the weights). We begin by focusing on enhanced risk-adjusted out-of-sample performance covariance parameter estimates and in a manner that is similar to some of the explain how to meet the main challenge aforementioned improved covariance matrix of sample risk reduction.2 Against this estimators. backdrop, we present the state-of-the art methods of reducing the problem In these papers, the focus was on testing of dimensionality and estimating the the out-of-sample performance of global covariance matrix with multi-factor models. minimum variance (GMV) portfolios, as We then turn to expected return estimation. opposed to the MSR portfolios (also known We argue that statistical methods are not as tangency portfolios), given that there is likely to generate any robust expected a consensus that purely statistical estimates return estimates, which suggests that of expected returns are not robust enough economic models such as the capital asset to be used, a point we return to later in this pricing model (CAPM) and the arbitrage paper when we look at expected return pricing theory (APT) should instead estimation. be used to estimate expected returns. Finally, we present evidence that proxies for14 An EDHEC-Risk Institute Publication
  15. 15. New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of thePerformance-Seeking PortfolioExhibit 1: Inefficiency of cap-weighted benchmarks and the quest for an efficient proxyfor the true tangency portfolio.The key problem in covariance matrix (excess) returns, and εt the Nx1 vectorestimation is the curse of dimensionality; containing the zero mean uncorrelatedwhen a large number of stocks are considered, residuals εi t.The covariance matrix for thethe number of parameters to estimate grows asset returns, implied by a factor model,exponentially, where the majority of them is given by:are pairwise correlations. Ω = β ⋅ ΣF ⋅ β T + ΣεTherefore, at the estimation stage, the where ΣF is the KxK covariance matrix ofchallenge is to reduce the number of the risk factors and Σε an NxN covariancefactors that come into play. In general, a matrix of the residuals corresponding tomultifactor model decomposes the (excess) each asset.return (in excess of the risk-free asset) of anasset into its expected rewards for exposure Although the factor-based estimator isto the “true” risk factors as follows: expected to allow for a reasonable trade- K off between sample risk and model risk, ri t = αi t + ∑ βi , j t ⋅F j t + εi t there is still the problem of choosing the j =1 “right” factor model. One popular approachor in matrix form for all N assets: attempts to rely as little as possible on rt = αt + βt Ft + εt strong theoretical assumptions by using principal component analysis (PCA) towhere βt is a NxK matrix containing determine the underlying risk factorsthe sensitivities of each asset i to the from the data. The PCA method is basedcorresponding j-th factor movements; rt is on a spectral decomposition of the samplethe vector of the N assets’ (excess) returns, covariance matrix and its goal is to use onlyFt a vector containing the K risk factors’ a few linear combinations of the original An EDHEC-Risk Institute Publication 15
  16. 16. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio stochastic variables—combinations that higher-order moments and co-moments of will constitute the set of (unobservable) the return distribution. This is a formidable factors—to explain covariance structures. challenge that greatly exacerbates the dimensionality problem already present Bengtsson and Holst (2002) and Fujiwara with mean-variance analysis. In a recent et al. (2006) motivate the use of PCA in a paper, Martellini and Ziemann (2010) similar way, extracting principal components extend the existing literature, which has to estimate expected correlation within focused mostly on the covariance matrix, MV portfolio optimisation. The latter find by introducing improved estimators for the that the realised risk-return of portfolios co-skewness and co-kurtosis parameters. based on the PCA method outperforms On the one hand, they find that the use the portfolio based on a single index and of these enhanced estimates generates that the optimisation gives a practically a significant improvement in investor reasonable asset allocation. On the whole, welfare. On the other hand, they find that the main strength of the PCA approach at when the number of constituents in the this point is that it leads to “letting the portfolios is large (more than twenty, for data talk” and having them tell us what the example), the increase in sample risk arising underlying risk factors that govern most of from the need to estimate higher-order the variability of the assets at each point in co-moments far outweighs the benefits time are. This strongly contrasts with having of considering a more general portfolio to rely on the assumption that a particular optimisation procedure. When portfolios factor model is the true pricing model and with large numbers of assets are optimised, reduces the specification risk embedded maximising the Sharpe ratio leads to better in the factor-based approach, all while out-of-sample results than does maximising keeping the sample risk reduction. a return-to-VaR ratio. It does so even when portfolio performance is assessed with The question of determining the appropriate measures that rely on VaR rather than on number of factors to structure the volatility to adjust for risk. Similar arguments correlation matrix is critical for the risk hold for other extreme risk measures such estimation when PCA is used as a factor as CVaR. In the end, using extreme risk model. Several options, some on greater measures in portfolios with large numbers theoretical grounds than others, have been of assets leads to a formidable estimation proposed to answer this question. problem, and empirical results suggest that it is sensible to stay with the mean-variance As a final note, we need to recognise approach, in which reliable input estimates that the discussion is so far cast in a can be derived. mean-variance setting, which can, in principle, be rationalised only for normally 1.1.2. Robust Estimators for Expected distributed asset returns. In the presence Returns of non-normally distributed asset returns, Although it appears that risk parameters optimal portfolio selection techniques can be estimated with a fair degree of require estimates for variance-covariance accuracy, expected returns are difficult to parameters, along with estimates for obtain with a reasonable estimation error16 An EDHEC-Risk Institute Publication
  17. 17. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio (Merton 1980). What makes the problem Taken together, these findings suggest worse is that optimisation techniques are that total risk, a model-free quantity given very sensitive to differences in expected by the sum of systematic and specific risk, returns, so portfolio optimisers usually should be positively related to expected allocate the largest fraction of capital to return. Most commonly, total risk is the the asset class for which estimation error in volatility of a stock’s returns. Martellini the expected returns is the largest (Britten- (2008) has investigated the portfolio Jones 1999; Michaud 1998). implications of these findings and found that tangency portfolios constructed on the In view of the difficulty of using sample- assumption that the cross-section of excess based expected return estimates in a expected returns could be approximated portfolio optimisation context, a reasonable by the cross-section of volatility posted alternative is to use some risk estimate better out-of-sample risk-adjusted as a proxy for excess expected returns.3 performance than their market-cap- This approach is based on the most basic weighted counterparts. principle in finance, i.e., the natural relationship between risk and reward. In More generally, recent research suggests3 - This discussion focuses onestimating the fair neutral fact, standard asset pricing theories such that the cross-section of expected returnsreward for holding risky as the APT imply that expected returns might best be explained by risk indicatorsassets. If one has access toactive views on expected should be positively related to systematic taking into account higher-order moments.returns, one may use adisciplined approach (e.g., the volatility, as measured through a factor Theoretical models have shown that, inBlack-Litterman model) to model that summarises individual stock exchange for higher skewness and lowercombine the active views andthe neutral estimates. return exposure to a number of rewarded kurtosis of returns, investors are willing to4 - For a similar conclusionfrom a behavioural risk factors. accept expected returns lower (and volatilityperspective, see Barberis and higher) than those of the mean-varianceHuang (2001). More recently, several papers have focused benchmark (Rubinstein 1973; Krauz and on the explanatory power of idiosyncratic Litzenberger 1976). More specifically, rather than systematic risk for the cross- skewness and kurtosis in individual stock section of expected returns. In particular, returns (as opposed to the skewness and Malkiel and Xu (2006), extending an kurtosis of aggregate portfolios) have been insight from Merton (1987), show that shown to matter in several papers. High an inability to hold the market portfolio, skewness is associated with lower expected whatever the cause, will force investors returns in several studies (Barberis and to deal, to some degree, with total risk Huang 2004; Brunnermeier, Gollier, and in addition to market risk, so firms with Parker 2007; Mitton and Vorkink 2007). The larger firm-specific variances require intuition behind this result is that investors higher average returns to compensate like to hold positively skewed portfolios. investors for holding imperfectly diversified The highest skewness is achieved by portfolios.4 That stocks with high concentrating portfolios in a small number idiosyncratic risk earn higher returns has of stocks that themselves have positively also been confirmed in a number of recent skewed returns. Thus investors tend to be empirical studies. underdiversified and drive up the price of stocks with high positive skewness, which in An EDHEC-Risk Institute Publication 17
  18. 18. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio turn reduces their future expected returns. that efficient equity benchmarks designed Stocks with negative skewness are relatively on the basis of robust estimates for risk and unattractive and thus have low prices and expected return parameters substantially high returns. The preference for kurtosis is in outperform in terms of risk-adjusted the sense that investors like low kurtosis and performance the market-cap-weighted thus expected returns should be positively indices often used as default options for related to kurtosis. Two studies provide investment benchmarks in spite of their empirical evidence that individual stocks’ well-documented lack of efficiency (Haugen skewness and kurtosis are indeed related to and Baker 1991; Grinold 1992). future returns (Boyer, Mitton, and Vorkink 2010; Conrad, Dittmar, and Ghysels 2008). Exhibit 2, borrowed from Amenc et al. (2010), An alternative to direct consideration shows summary performance statistics for of the higher moments of returns is to an efficient index constructed in keeping use a risk measure that aggregates the with the aforementioned principles. For dimensions of risk. In this respect, Bali the average return, volatility, and Sharpe and Cakici (2004) show that future returns ratio, we report differences with respect on stocks are positively related to their to cap-weighting and assess whether this Value-at-Risk and Estrada (2000) and Chen, difference is statistically significant. Chen, and Chen (2009) show that there is a relationship between downside risk and Exhibit 2 shows that the efficient weighting expected returns. of index constituents leads to higher average returns, lower volatility, and higher Sharpe 1.1.3. Implications for Benchmark ratios. All these differences are statistically Portfolio Construction significant at the 10% level, whereas the Once careful estimates for risk and return difference in Sharpe ratios is significant parameters have been obtained, one may even at the 0.1% level. Given the data, it is then design efficient proxies for asset class highly unlikely that the unobservable true benchmarks with attractive risk/return performance of efficient weighting was profiles. For example Amenc et al. (2010) find not different from that of capitalisation Exhibit 2: Risk and return characteristics for the efficient index Index Ann. average Ann. standard Sharpe ratio Information Tracking return deviation (compounded) ratio error (compounded) Efficient index 11.63% 14.65% 0.41 0.52 4.65% Cap-weighted 9.23% 15.20% 0.24 0.00 0.00% Difference 2.40% -0.55% 0.17 - - (efficient minus cap-weighted) p-value for difference 0.14% 6.04% 0.04% - - The table shows risk and return statistics portfolios constructed with the same set of constituents as the cap-weighted index. Rebalancing is quarterly subject to an optimal control of portfolio turnover (by setting the reoptimisation threshold to 50%). Portfolios are constructed by maximising the Sharpe ratio given an expected return estimate and a covariance estimate. The expected return estimate is set to the median total risk of stocks in the same decile when sorting by total risk. An implicit factor model for stock returns is used to estimate the covariance matrix. Weight constraints are set so that each stocks weight is between 1/2N and 2/N, where N is the number of index constituents. P-values for differences are computed using the paired t-test for the average, the F-test for volatility, and a Jobson-Korkie test for the Sharpe ratio. The results are based on weekly return data from 01/1959 to 12/2008.18 An EDHEC-Risk Institute Publication
  19. 19. New Frontiers in Benchmarking and Liability-Driven Investing - September 20101. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of thePerformance-Seeking Portfolioweighting. Economically, the performance a factor model approach (Ledoit and Wolfdifference is pronounced, as the Sharpe 2003; Ledoit and Wolf 2004).ratio increases by about 70%.1.2. Asset Allocation Step: Puttingthe Efficient Benchmarks TogetherAfter efficient benchmarks have beendesigned for various asset classes, thesebuilding blocks can be assembled in a secondstep, the asset allocation step, to build awell-designed multiclass performance-seeking portfolio. Although the methodswe have discussed so far can, in principle,be applied in both contexts, a number ofkey differences should be emphasised.In the asset allocation context, the numberof constituents is small, and using time-and state-dependent covariance matrixestimates is reasonable; nonetheless, theseestimates do not necessarily improve thesituation in portfolio construction contexts,in which the number of constituents islarge. Similarly, although it is not, ingeneral, feasible to optimise the portfoliowith higher-order moments in a portfolioconstruction context, in which the numberof constituents is typically large, it isreasonable to go beyond mean-varianceanalysis in an asset allocation context,in which the number of constituents islimited.Furthermore, in asset allocation, the universeis not homogeneous, which has implicationsfor expected returns and covarianceestimation. In terms of covariance matrix,it will not prove easy to obtain a universalfactor model for the entire investmentuniverse. In this context, it is arguably betterto use statistical shrinkage towards, say, theconstant correlation model, than to take An EDHEC-Risk Institute Publication 19
  20. 20. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 1. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Performance-Seeking Portfolio20 An EDHEC-Risk Institute Publication
  21. 21. 2. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Liability-Hedging Portfolio An EDHEC-Risk Institute Publication 21
  22. 22. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 2. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Liability-Hedging Portfolio Risk diversification is only one possible of cash outflows to be paid, the real value form of risk management, focusing merely of which is known today, but for which the on achieving the best risk/return trade-off nominal value is typically matched with an regardless of investment objectives and inflation index. It is possible, in theory, to constraints. On the other hand, one should construct a portfolio of assets whose future recognise that diversification is simply cash flows will be identical to this structure not the appropriate tool when it comes to of commitments. Doing so would involve protecting long-term liability needs. purchasing—assuming that securities of that kind exist on the market—inflation- One key academic insight from the linked zero-coupon bonds with a maturity pioneering work of Robert Merton in the corresponding to the dates on which the nineteen-seventies is that the presence of monthly pension instalments are paid out, state variables impacting the asset return with amounts that are proportional to the and/or wealth process will lead to the amount of real commitments. introduction of dedicated hedging demands, in addition to cash and optimally diversified Although this technique has the advantage PSP (which is still needed). of simplicity and, in theory, allows perfect risk management, it has a number of In particular, it is clear that the risk factors limitations and implementation poses impacting pension liability values should be several challenges. In particular, finding hedged rather than diversified away. Two of bond portfolios with the proper duration is these factors, interest rate risk and inflation hardly feasible, especially in the corporate risk, stand out. Although constructing bond segment. interest rate and inflation hedging benchmarks might seem straightforward The conflicts of interest between issuers and compared to constructing performance- investors about the duration of corporate seeking benchmarks, some challenges bonds is known as the duration problem. remain, which we discuss now. Each bond investor has in mind a specific time horizon, and there is no reason to expect that these needs correspond to the 2.1. Towards the Design of Improved optimal financing plan of the issuers. In Interest Rate Risk Benchmarks fact, the duration structure of outstanding A first approach to the design of the bonds reflects the preferences of the issuers LHP, called cash-flow matching, involves in their aim to minimise the cost of capital. ensuring a perfect static match between This minimisation is fundamentally opposed the cash flows from the portfolio of assets to the interest of the investors, who usually and the commitments in the liabilities. Let try to maximise their returns. Although us assume, for example, that a pension fund as such a part of the suitability problem has a commitment to pay out a monthly mentioned above, the duration mismatch in pension to a retired person. Leaving aside the corporate bond market is of primordial the complexity relating to the uncertain life importance to investors. Pension funds have expectancy of the retiree, the structure of some fixed nominal liabilities originating the liabilities is defined simply as a series from their defined-benefit plans. Given this22 An EDHEC-Risk Institute Publication
  23. 23. New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of theLiability-Hedging Portfoliolong-term perspective, long-term bonds liability-matching portfolios, the soleare a much better hedge than short-term purpose of which is to hedge away asdebt. Issuers of such bonds therefore have effectively as possible the impact ofto pay only a small yield premium—even unexpected changes in the risk factors—though they are more volatile. In contrast, most notably inflation—affecting liabilityfor short-term investors with no fixed time values. A variety of cash instrumentshorizon in mind such investments are far (Treasury inflation protected securities, orless attractive. The duration of the indices TIPS) as well as dedicated OTC derivativesis nonetheless a result of the sell-side of (such as inflation swaps) are used tocorporate bonds—so no investor should hold achieve a customised exposure to consumerjust this benchmark duration. Hence, many price inflation. One outstanding problem,corporate bond indices are not well suited however, is that such solutions generateto serving as benchmarks for corporate very modest performance given that realbond investors. returns on inflation-protected securities, negatively impacted by the presence ofMore worrisome perhaps is that the a significant inflation risk premium, arecharacteristics of corporate bond indices usually very low. In this context, it hascan change over time. So, efforts to design been argued that some other asset classes,stable corporate bond indices optimised such as stocks, real estate, or commodities,in an attempt not only to maximise their could provide useful inflation protection,risk-adjusted performance but also to especially when long-term horizons aredisplay a (quasi-) constant duration and considered, at a cost lower than that ofallocation by rating class over time are investing in TIPS.required. Empirical evidence suggests that there is in fact a negative relationship between2.2. Towards the Design of Improved expected stock returns and expectedInflation-Hedging Benchmarks inflation, which is consistent with theA recent surge in worldwide inflation has intuition that higher inflation depressesincreased the need for investors to hedge economic activity and thus depresses stockagainst unexpected changes in prices. returns. On the other hand, higher futureInflation hedging has in fact become a inflation leads to higher dividends andconcern of critical importance for private thus higher returns on stocks, so equityinvestors, who consider inflation a direct investments should offer significantthreat to their purchasing power, as well as inflation protection over long horizons (asfor pension funds, which must make pension it happens, several recent empirical studiespayments often indexed to consumer prices [Boudoukh and Richardson 1993; Schotmanor wages. and Schweizer 2000] have confirmed that equities provide a good hedge againstIn this context, novel forms of institutional inflation over the long term). This propertyinvestment solutions have been promoted is particularly appealing for long-termby asset managers and investment banks, investors such as pension funds, which needfocusing on the design of customised to match price increases at the horizon but An EDHEC-Risk Institute Publication 23
  24. 24. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 2. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Liability-Hedging Portfolio not on a monthly basis. Obviously, different Commodity prices, in particular, are believed kinds of stocks offer different inflation- to be leading indicators of inflation in that hedging benefits, and it is in fact possible they are quick to respond to economy-wide to select stocks or sectors on the basis of shocks to demand. Commodity prices are their ability to hedge against inflation. usually set in highly competitive auction For example, utilities and infrastructure markets and consequently tend to be more typically have revenues highly correlated flexible than prices in general. In addition, with inflation, and as a result they tend recent inflation is fuelled heavily by to provide better-than-average inflation increases in commodity prices, in particular protection. So it seems possible to select in agriculture, minerals, and energy. In the stocks or sectors on the basis of their same vein, commercial and residential ability to hedge against inflation (hedging real estate provide at least a partial hedge demand), as opposed to selecting them as a against inflation, and portfolios that function of their outperformance potential include real estate realise an increase in (speculative demand). In this context, one inflation hedgeability, especially over longer can envision selecting stocks or sectors in an horizons. attempt to maximise the inflation-hedging property of equity-based inflation-hedging Exhibit 3 (taken from Amenc, Martellini, solutions. The analysis typically involves two and Ziemann [2009], a paper to which we separate phases, selection and optimisation. refer for further details on the calibration The goal of the selection phase is to select of the VAR and VECM models), which the set of stocks likely to exhibit the most displays a set of estimated term structure attractive inflation-hedging properties. In of correlation coefficients between asset the second phase, a portfolio of selected returns and inflation-linked liability returns. stocks will be formed in such a way as to It confirms that various asset classes have optimise the expected inflation-hedging different inflation-hedging properties over benefits. various horizons; inflation-hedging capacity increases in tandem with the horizon for Going beyond the equity universe, similar stocks, bonds, and real estate. inflation-hedging properties are expected for bond returns. Indeed, bond yields may As a consequence of the aforementioned be decomposed into a real yield and an findings, it is tempting to investigate expected inflation component. Since whether novel liability-hedging investments expected and realised inflation move can be designed to decrease the cost to the together over the long term, a positive investor of inflation insurance. In particular, long-term correlation between bond returns it is possible to construct different versions and changes in inflation is expected. In the of the inflation-hedging portfolio to assess short-term, however, expected inflation the impact of introducing investment classes may deviate from actual realised inflation, such as equities, commodities, and real leading to low or negative correlations in estate in addition to inflation-linked bonds. the short term. It has also been recently Amenc, Martellini, and Ziemann (2009) argued that alternative forms of investment have shown that the increased expected offer attractive inflation-hedging benefits. return generated by adding asset classes24 An EDHEC-Risk Institute Publication
  25. 25. New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of theLiability-Hedging PortfolioExhibit 3: Term structure of correlation coefficients between different asset returns and inflation-linked liability returns for varioushorizonsDotted lines correspond to implied correlations estimated from a vector auto regressive (VAR) model, whereas solid lines correspondto implied correlations estimated from a vector error correction model (VECM).Source: Amenc, Martellini, and Ziemann (2009).with good long-term inflation-hedging and hence enhance the performance ofproperties allows pension fund sponsors to the inflation-hedging portfolio.maintain contributions and lower exposureto downside risk. As a final note, we analyse in the next section situations in which the separationOther advanced solutions may involve of the performance-seeking and liability-hedging a particular segment of inflation hedging portfolios does not apply in adistribution, with an expected focus on strict sense. More specifically, we considerhedging large, as opposed to moderate, situations in which there are multiple andinflation shocks, in an attempt to reduce equally attractive candidates for the PSPyet again the costs of inflation hedging and LHP. An EDHEC-Risk Institute Publication 25
  26. 26. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 2. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Liability-Hedging Portfolio 2.3.Performance-Seeking Portfolios If an investor were given a choice of two with Attractive Liability- /Inflation- (or more) performance-seeking portfolios Hedging Properties with identical risk/reward ratios but distinct As previously mentioned, asset pricing liability-hedging properties he would theory is grounded on separation theorems obviously favour the performance portfolio that state that risk and performance are with the most attractive liability-hedging two conflicting objectives best managed properties. separately. According to this paradigm, performance generation is obtained first In fact, this hypothetical question would through optimal exposure to rewarded not arise if the efficient frontier is strictly risk factors to alleviate the burden on concave, which would ensure the existence contributions, while hedging against and uniqueness of the maximum risk/ unexpected shocks that impact current reward ratio portfolio, as is the case in the value of (assets and) liabilities is accounted standard mean-variance paradigm with for by a separate dedicated portfolio. perfect information. It has been shown, however, that when a This clear separation of performance and general, not strictly convex risk measure is hedging portfolios is very useful and has used the efficient frontier may not be strictly a number of important implications not concave, and as a result the max reward/ only for portfolio construction and asset risk portfolio may not be unique (Stoyanov, allocation techniques but also for the Rachev, and Fabozzi 2007). A similar result organisational structure of the institutional would hold for a mean-variance objective investor. in the absence of perfect information regarding the risk/return parameters. See exhibit 4 for an illustration. As a result, in Exhibit 4: Multiple true or estimated tangency portfolios26 An EDHEC-Risk Institute Publication
  27. 27. New Frontiers in Benchmarking and Liability-Driven Investing - September 20102. Asset Allocation and Portfolio ConstructionDecisions in the Optimal Design of theLiability-Hedging Portfoliomost realistic situations, if given a choiceof seemingly attractive portfolio-seekingportfolios, it would be rational for aninvestor to select the performance portfoliowith the most attractive liability-hedgingproperties, and this would not conflict withthe fund separation theorem. Conversely, ifan investor had a choice of liability-hedgingportfolios with equally attractive inflation-hedging benefits, the investor would tendto favour the hedging portfolio with themost attractive risk/reward ratio. An EDHEC-Risk Institute Publication 27
  28. 28. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 2. Asset Allocation and Portfolio Construction Decisions in the Optimal Design of the Liability-Hedging Portfolio28 An EDHEC-Risk Institute Publication
  29. 29. 3. Dynamic Allocation Decisionsto the Performance-Seeking and Liability-Hedging Portfolios An EDHEC-Risk Institute Publication 29
  30. 30. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 3. Dynamic Allocation Decisions to the Performance-Seeking and Liability-Hedging Portfolios Assuming that reasonable proxies for already the case in equation (2).6 The performance-seeking and liability-hedging allocation to the “safe” building block is portfolios have been designed using an increasing function of the beta β of some of the aforementioned methods, liability portfolio with respect to the we must still determine the optimal liability-hedging portfolio. If there is an strategy to allocate assets to those two asset portfolio that perfectly matches building blocks. Several novel paradigms, the liability value, then the beta is 1, and which we describe next, are reshaping an infinitely risk-averse investor fully our approaches to long-term investment allocates to the LHP. This is consistent with decisions for long-term investors facing the intuition that for an investor facing liability commitments and short-term liability commitments the LHP, as opposed performance constraints. to cash, is the true risk-free asset. Equation (2) is the solution to a 3.1. Accounting for the Presence of static optimisation problem, and the Investors’ Liability Commitments: corresponding strategy is (by design) of The Liability-Driven Investment the buy-and-hold kind. Equation (3) is 5 - See appendix A.2 for details. Paradigm the solution to a dynamic optimisation 6 - Risk aversion is not As explained earlier in this paper, investors problem, as shall be evidenced by the observable, and not even well-defined for institutions, with consumption/liability objectives presence of an explicit time dependency in but it should be treated as an implicit parameter that can need to invest in two distinct portfolios, the expression for the optimal allocation be inferred from a given risk in addition to cash: one performance- strategy. The corresponding strategy is a budget, often expressed in terms of expected shortfall seeking portfolio and one liability-hedging fixed-mix strategy, where, in principle, with respect to liabilities. 7 - For more details, see portfolio, construction methods for which constant trading occurs to rebalance the Martellini and Milhau (2009). have been discussed in previous sections. portfolio allocation back to the constant target. The fund separation theorem is Formally, under the assumption of a expressed here under the assumption of a constant opportunity set,5 we obtain constant opportunity set. In later sections, the following expression of the fund we shall relax this assumption and analyse separation theorem in the intertemporal how the allocation is impacted by the context when trading is possible between introduction of time variation to the current date and investment horizon: expected return and volatility of the PSP. λ ⎛ 1⎞ wt* = PSP + ⎜ 1− ⎟ β LHP (3) γσ ⎝ γ⎠ Exhibit 5 shows the distribution of the funding ratio, defined as asset value This expression is similar to that in divided by liability value, at horizon. The equation (2), extended to the asset/ initial funding ratio is assumed to be liability management setting. As appears 100%; the horizon is 11.32 years (taken to from equation (3), the allocation to the be the duration of liabilities of a Dutch “risky” building block is still an increasing defined-benefit pension fund.7 On the function of the PSP Sharpe ratio λ and a left-hand side the Sharpe ratio of the decreasing function of the investor’s risk PSP is assumed to be 0.24, whereas it is aversion γ and PSP volatility σ, as was assumed to have improved by 50% to30 An EDHEC-Risk Institute Publication
  31. 31. New Frontiers in Benchmarking and Liability-Driven Investing - September 20103. Dynamic Allocation Decisions to thePerformance-Seeking and Liability-HedgingPortfolios0.36 on the right-hand side. The idea 3.2. Accounting for the Presencehere is to capture possible improvements of Investors’ Long-Term Objectives:to the performance-seeking Sharpe ratio The Life-Cycle Investment Paradigmthat would result from the use of efficient Although it may be acceptable to assumerather than cap-weighted benchmarks, as a constant opportunity set when investorsdiscussed in previous sections. have a short-term horizon, a long horizon, typical of most investors’ problems, makesThe expected funding ratio has increased it necessary to go beyond Markowitzsignificantly. Volatility increases as well, static portfolio selection analysis. Thebut this is mostly the result of higher next important step after Markowitzdispersion on the upside. Regarding (1952) is Merton (1969, 1971), whodownside risk, the shortfall probability takes portfolio construction techniques(formally defined as the probability that beyond the static setting and shows howthe funding ratio ends up below 100%) to use dynamic programming to solvedecreases from 19.23% to 11.97% when dynamic portfolio optimisation problems.the Sharpe ratio of the PSP portfolio In terms of industry implications, therises from 0.24 to 0.36. On the whole, development of dynamic asset pricingthe substantial improvement in the theory has led to the emergence ofdistribution of the funding ratio at horizon improved investment solutions that takeis the result of two effects. On the one into account the changing nature ofhand, if the Sharpe ratio of the PSP rises, investment opportunities. These novelthe expected value of the funding ratio forms of investment are broadly referredimproves, with associated benefits from an to as life-cycle investing strategies.ALM perspective. On the other hand, theallocation to the PSP has also increased. Current forms of life-cycle investingOverall, we obtain that improving the are sometimes grossly sub-optimal.risk/reward ratio is a key component in For example, a popular asset allocationmeeting investors’ long-term objectives. strategy for managing equity risk onExhibit 5: Distribution of funding ratio with inefficient versus efficient PSPThe initial funding ratio is assumed to be 100%; the horizon is 11.32 years (taken to be the duration of liabilities of a Dutch defined-benefit pension fund). On the left-hand side the Sharpe ratio of the PSP is assumed to be 0.24, whereas it is assumed to haveimproved by 50% to 0.36 on the right-hand side. An EDHEC-Risk Institute Publication 31
  32. 32. New Frontiers in Benchmarking and Liability-Driven Investing - September 2010 3. Dynamic Allocation Decisions to the Performance-Seeking and Liability-Hedging Portfolios behalf of a private investor in the context in these variables have an impact on of a defined-contribution pension plan is portfolio risk and performance (through known as deterministic life-cycle investing. changes in interest rates and risk premium In the early stages, when the retirement process parameters), which should be date is far away, the contributions managed optimally (Detemple and are invested entirely in equities. Then, Rindisbacher 2009). In fact, one can show beginning on a predetermined date (ten that the dynamic asset allocation problem years, say) before retirement, the assets involves an optimal hedging of the state are switched gradually to bonds at some variables impacting the risk-free rate and pre-defined rate (10% a year, say). By the the risk-premium processes. Except for date of retirement, all the assets are held a myopic investor, the optimal dynamic in bonds.8 This is somewhat reminiscent solution does not consist of merely taking of the rule of thumb put forward by the static solution and updating it with Shiller (2005), advocating a percentage time-varying parameters; one should also allocation to equity given by 100 minus introduce dedicated hedging portfolios. the investor’s age in years. For example, when interest rates are 8 - In fact, the rationale behind the strategy is to While deterministic life-cycle investing is stochastic, cash is no longer a risk-free reduce the impact on the a simple strategy popular with investment asset; the risk-free asset is instead a bond pension of a catastrophic fall in the stock market just managers and consultants, and it is widely with a term to maturity matching the before the plan member retires and to hedge the used by defined-contribution pension investor’s horizon. It is hardly surprising, interest-rate risk inherent to providers, there is no evidence that it is an then, that the optimal allocation decision the pension-related liability value. As we will argue later, optimal strategy in a rational sense, and in the presence of interest rate risk holding no exposure to equity is a very trivial way we argue below that these strategies are involves an additional building block, the of managing equity risk, and very imperfect proxies for truly optimal bond with a term to maturity matching one that keeps investors from enjoying equity upside stochastic life-cycle investing strategies. the investor’s horizon, in addition to cash potential. In general, the presence of risk factors and the highest risk/reward performance- can impact the opportunity set (risk and seeking portfolio (a three-fund rather return parameters), and they can have a than two-fund separation theorem). As direct impact on the wealth process for another illustration, let us assume that non-self-financed portfolios when inflows the expected return on most risky assets and outflows of cash are taking place. is, for example, negatively impacted by increases in oil prices. To compensate 3.2.1. Accounting for Risk Factors for the deterioration of the investment Impacting the Investment Opportunity opportunity set in the event of a sharp Set increase in oil prices, the investor will A large body of empirical research has benefit from holding a long position in a shown that interest and inflation rates, portfolio optimised to exhibit the highest as well as expected return, volatility, and possible correlation with oil prices. correlation parameters are stochastically time-varying, as a function of key state Kim and Omberg (1996), for example, have variables that describe the state of the analysed a model including a stochastic business cycle. Unexpected changes equity risk premium with a mean-32 An EDHEC-Risk Institute Publication
  33. 33. New Frontiers in Benchmarking and Liability-Driven Investing - September 20103. Dynamic Allocation Decisions to thePerformance-Seeking and Liability-HedgingPortfoliosreverting component. In this model, one than when it is constant (σλ=0);can show that the optimal allocation • The hedging demand disappears if there involves not only a deterministic decrease is no equity risk premium risk (σλ=0), or ifof the allocation to equity as the investor the risk exists but cannot be hedged awaygets closer to the time horizon, which (ρλS=0).is consistent with standard target • The investment in stock decreases when date fund practice, but also a state- approaching horizon T; this is consistentdependent component, suggesting with the prescriptions of target datethat the allocation to equity should be funds.increased (respectively, decreased) when,as measured by such proxies as dividend We also confirm that there is oneyields or price-earnings ratios, equity has additional state-dependent factor: if/become cheap and decreased when it has when equity prices are low (high), andbecome expensive. therefore expected return is high (low), one should allocate more (less) to stocks,We obtain the following expression for the regardless of horizon.optimal allocation strategy, assuming forsimplicity that an equity benchmark is the 3.2.2. Accounting for Risk Factorsonly risky asset so that the PSP is 100% Impacting the Wealth Processinvested in that benchmark (see appendix As noted earlier, the presence of riskA.3 for details): factors can also have a direct impact on the wealth process, even in the case of a constant opportunity set. In fact, most portfolios are not self-financed portfolios, because of the presence of outflows of cash (consumption, liability payments)Here we let ρλS be the correlation of and/or inflows of cash (endowment,the Sharpe ratio of the equity index, contribution, income, and so on). Fordenoted by λS, and the equity index example, labour income risk will havereturn; a negative correlation means that an impact on optimal portfolio decisionshigh realised return periods tend to be and will legitimise the introduction offollowed by low expected return periods, a dedicated hedging demand. Morewhich is supported by empirical evidence. generally, the present value of liabilityMoreover, σλ is the volatility of the equity and endowment flows are impacted bySharpe ratio process and σS is the volatility state variables (interest rate risk, inflationon the stock index. risk, income risk, etc.), the impact of which must be hedged away.The hedging demand againstunexpected changes in the PSP Sharpe A pension fund, for example, should holdratio has the following properties (for ρλ S<0 a long position in a liability-hedgingand γ>1): portfolio to hedge away the implicit short• The investor with γ>1 holds more stocks position in liability flows. Conversely, awhen equity Sharpe ratio is mean-reverting sovereign wealth fund from an oil-rich An EDHEC-Risk Institute Publication 33

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