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- 1. An EDHEC Risk and Asset Management Research Centre Publication Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management January 2009 Sponsored by
- 2. Table of Contents Foreword ...........................................................................................................................2 Abstract ..............................................................................................................................5 1. Introduction ........................................................................................... 7 2. Modelling Return and Inflation Dynamics .........................................13 3. Data and Model Specification ...........................................................19 4. Inflation-Hedging Properties of Various Assets and Portfolios ...25 5. Implication for Risk Budgeting in Asset-Liability Management .. 31 6. Conclusions and Directions for Future Research ............................35 Appendix ........................................................................................................................ 39 References ..................................................................................................................... 59 About the EDHEC Risk and Asset Management Research Centre .............. 64 About Morgan Stanley Investment Management ........................................... 69 Printed in France, January 2009. Copyright EDHEC 2009. The opinions expressed in this survey are those of the authors and do not necessarily reflect those of EDHEC Business School.
- 3. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Foreword The present publication is the first to be I would like to thank my co-authors, drawn from the EDHEC/Morgan Stanley Lionel Martellini and Volker Ziemann, for Investment Management research chair the extensive work that they have done in on Financial Engineering and Global this area. We hope that you will find our Alternative Portfolios for Institutional analysis and conclusions interesting and Investors. will continue to monitor and contribute to our research in this area. This chair, which corresponds to a three- year partnership between Morgan Stanley Finally, we would like to extend our Investment Management and the EDHEC warmest thanks to our partners at Morgan Risk and Asset Management Research Stanley Investment Management for Centre, adapts risk budgeting and risk their commitment to this research chair management concepts and techniques to and their close involvement with our the specificities of alternative investments, research. both in the context of asset management and asset-liability management. The Wishing you an enjoyable and informative research is led by Professor Lionel read, Martellini, Scientific Director of the EDHEC Risk and Asset Management Research Centre. In the current context of a short- and Noël Amenc probably medium-term downturn in the Professor of Finance Director of the EDHEC Risk and Asset Management financial markets, our research seems to Research Centre be of particular relevance for long-term investors. Our results suggest that real estate and commodities have particularly attractive inflation hedging properties over long-horizons, which justify their introduction in pension funds' liability- matching portfolios. We show that novel liability-hedging investment solutions, including commodities and real estate in addition to inflation-linked securities, can be designed so as to decrease the cost of inflation insurance for long-horizon investors. An EDHEC Risk and Asset Management Research Centre Publication 3
- 4. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 About the Authors Noël Amenc is Professor of finance and Director of Research and Development at EDHEC Business School, where he heads the Risk and Asset Management Research Centre. He has a Masters degree 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 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 the EDHEC Risk and Asset Management Research Centre. Lionel has consulted on risk management, alternative investment strategies, and performance benchmarks for various institutional investors, investment banks, and asset management firms, both in Europe and in the United States. His research has been published in leading academic and practitioner journals, including the Journal of Mathematical Economics, Management Science, Review of Financial Studies, the Journal of Portfolio Management, Financial Analysts Journal, and Risk. He sits on the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. Lionel has co-authored and co-edited reference texts on fixed-income management and alternative investment such as the much-praised Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies (Wiley Finance) and is regularly invited to deliver presentations at leading academic and industry conferences. He holds graduate degrees in business administration, economics, statistics, and mathematics, as well as a PhD in finance from the Haas School of Business at UC Berkeley. Volker Ziemann is a Senior Research Engineer at EDHEC Risk and Asset Management Research Centre. He holds Master’s degrees in economics and statistics and a PhD in finance. His research focus is on the econometrics of financial markets and optimal asset allocation decisions involving non-linear payoffs. 4 An EDHEC Risk and Asset Management Research Centre Publication
- 5. Abstract A n E D H E C R i s k a n d A s s e t M a n a g e m e n t R e s e a rc h C e n tre Pub l i ca ti on 5
- 6. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Abstract This paper presents an empirical analysis of the benefits of alternative forms of investment strategies from an asset- liability management perspective. Using a vector error correction model (VECM) that explicitly distinguishes between short- term and long-term dynamics in the joint distribution of asset returns and inflation, we identify the presence of long-term cointegration relationships between the return on typical pension fund liabilities and the return of various traditional and alternative asset classes. Our results suggest that real estate and commodities have particularly attractive inflation hedging properties over long-horizons, which justify their introduction in pension funds' liability-matching portfolios. We show that novel liability-hedging investment solutions, including commodities and real estate in addition to inflation-linked securities, can be designed so as to decrease the cost of inflation insurance for long-horizon investors. These solutions are shown to achieve satisfactory levels of inflation hedging over the long-term at a lower cost compared to a solution solely based on treasury inflation-protected securities (TIPS) or inflation swaps. Overall our results suggest that alternatives are very useful ingredients for institutional investors facing inflation-related liability constraints. 6 An EDHEC Risk and Asset Management Research Centre Publication
- 7. 1. Introduction A n E D H E C R i s k a n d A s s e t M a n a g e m e n t R e s e a rc h C e n tre Pub l i ca ti on 7
- 8. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Introduction A recent surge in worldwide inflation liability-hedging portfolio (LHP), the has increased the need for investors to sole purpose of which is to hedge away hedge against unexpected changes in price as effectively as possible the impact levels. In the most recent forecasts, global of unexpected changes in risk factors consumer price index (CPI) inflation has affecting liability values, and most been revised up more than a percentage notably interest rate and inflation risks. point, to 5.0%, for calendar year 2008, and This LHP complements the traditional by 0.7 percentage points, to 3.7%, in 2009, performance-seeking portfolio (PSP), the a trend that, given the long-term increased composition of which is not impacted pressure on food and energy resources, is by the presence of liabilities. Within likely to continue, despite the current crisis. the aforementioned LDI paradigm, a The CPI inflation peak has been forecasted variety of cash instruments (treasury to be 4.1% in the euro area and 5.5% in the inflation-protected securities, or TIPS) US, and the direct impact of higher food as well as dedicated OTC derivatives and energy costs is estimated to account (such as inflation swaps) are typically for 2.5 percentage points and 3 percentage used to tailor customised inflation points, respectively, of these rates.1 exposures that are suited to particular 1 - See Barclays Capital Inflation acceleration in emerging markets institutional investor’s liability profile. Global Outlook 2008, Implications for financial will likely be greater than it is in developed One outstanding problem, however, is markets. countries, a reflection of the much greater that such solutions generate very modest importance of food in the basket of performance. In fact, real returns on consumer goods in these markets and of inflation-protected securities, negatively their generally higher rates of growth. As impacted by the presence of a significant a result of these trends, inflation hedging inflation risk premium, are usually very is now of critical importance not only for low. In other words, while these solutions private investors, who consider inflation offer substantial risk management benefits, a direct threat to their purchasing power, the lack of performance makes them but also, and perhaps more importantly, costly options for pension funds and for pension funds, which may have their sponsors. In addition, the inflation- to make pension payments that are linked securities market does not have the indexed (conditional or full indexation) capacity to meet the collective demand to consumer price or wage indices. of institutional and private investors, while the OTC inflation derivatives market This focus on inflation hedging is suffers from a perceived increase in consistent with the heightened focus counterparty risk. on liability-risk management that has emerged as a consequence of the In this context, it has been argued that 2000-2003 pension crisis. A number of other asset classes, such as stocks and so-called liability-driven investment (LDI) nominal bonds, as well as real estate or techniques have been promoted over the commodities, could provide useful, albeit past few years by several investment imperfect, inflation protection at a lower banks and asset management firms, which cost than TIPS do. In the short term, equity advocate the design of a customised investments are relatively poor inflation 8 An EDHEC Risk and Asset Management Research Centre Publication
- 9. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Introduction hedging vehicles. Indeed, empirical bond returns and changes in inflation. evidence suggests that there is in fact a In the short-term, however, expected negative relationship between expected inflation may deviate from actual realised stock returns and expected inflation inflation, leading to low or negative (Fama and Schwert 1977; Gultekin correlation. There again, an investor willing 1983; Kaul 1987), which is consistent and able to relax short-term constraints with the intuition that higher inflation to focus on long-term inflation hedging leads to less economic activity and thus properties will find that investing in depresses stock returns (Fama 1981).2 nominal bonds can provide a cost-efficient Higher future inflation, however, alternative (or complement) to investing leads to higher dividends and thus higher in inflation-linked securities. returns on stocks (Campbell and Shiller 1988), so equity investments should Moving beyond traditional investment offer significant inflation protection over vehicles such as stocks and bonds, recent longer horizons, as has been confirmed academic research has also suggested by a number of recent empirical academic that alternative forms of investment studies (Boudoukh and Richardson offer attractive inflation-hedging benefits. 2 - This finding is also consistent with the so-called 1993; Schotman and Schweitzer Commodity prices, in particular, have "Fed model", which is used by 2000). This property is particularly been found to be leading indicators of many investment professionals to generate appealing for long-term investors such inflation in that they are quick to respond signals about the relative attractiveness of stock prices as pension funds, which need to match to economy-wide shocks to demand. relative to bond prices. increases in prices at the horizon, but Commodity prices are generally set in The model assumes that bonds and equities compete not necessarily on a monthly basis. highly competitive auction markets and for space in investment portfolios; if bond yields Obviously, different kinds of stocks offer consequently tend to be more flexible than increase, then stock yields different inflation-hedging benefits, and prices in general. Moreover, the rise in must also rise in order to remain competitive. Thus, the it is in fact possible to select stocks or the prices for agricultural, mineral and Fed model relates the yield on stocks (as measured by the sectors on the basis of their ability to energy commodities has been behind ratio of dividends or earnings hedge against inflation (hedging demand), much of the recent rise in inflation. to stock prices) to the yield on Treasury bonds and to as opposed to selecting them for their Gorton and Rouwenhorst (2006) find the relative risk premium of stocks versus bonds. In outperformance potential (speculative that, over the 1959-2004 period, the long-run, the Fed model demand). For example, utilities and commodity futures were positively posits that the actual yield on stocks will revert to a infrastructure companies usually have correlated with inflation, unexpected normal yield level given by the bond yield plus the risk revenues that are heavily correlated with inflation, and changes in expected premium. Historically, the inflation, and as a result they tend to provide inflation. They also find that inflation rate of inflation has been a major influence on better-than-average inflation protection. correlations tend to increase with the nominal bond yields. Therefore, the Fed model Similar inflation-hedging properties are holding period and are larger at return implies that stock yields and expected for bond returns. Indeed, bond intervals of one and five years than at inflation must be highly correlated. Campbell and yields may be decomposed into a real yield the monthly or quarterly frequency. Vuolteenaho (2004) review stock market performance and an expected inflation component. In the same spirit, it has also been found between 1927 and 2002, Since expected and realised inflation move that commercial and residential real estate examining the impact of risk premiums and inflation on together over the long term (Schotman provide at least a partial hedge against stock yields and find strong support for the Fed model. and Schweitzer 2000), we expect a inflation, which implies that portfolios positive long-term correlation between that include real estate allow enhanced An EDHEC Risk and Asset Management Research Centre Publication 9
- 10. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Introduction inflation-hedging benefits (Fama and returns with a vector auto-regressive (VAR) Schwert 1977, Hartzell et al. 1987; model on log-returns, as was common in Rubens et al. 1989). This effect seems previous research on the subject, omits to be particularly significant over long any information on price dependencies horizons. Hence, Anari and Kolari (2002) and long-term equilibria to focus purely examine the long-run impact of inflation on short-term effects in return series. on homeowner equity by investigating In order to address this shortcoming of the relationship between house prices the VAR model, we include cointegration and the prices of non-housing goods and relationships in the model and assess services, rather than return series and sensitivities of model-implied dynamics inflation rates, and infer that house prices with respect to these additional factors are a stable inflation hedge in the long that capture price dependencies in addition run. When it comes to securitised forms to return dependencies. The resulting error of real estate investment, the situation is correction form of the vector-autoregressive less clear, even though there is evidence model (VECM), or cointegrated VAR model, of a long-term connection that suggests has the striking advantage, as compared that REITs could neutralise part of the to the standard VAR representation, that inflation risk in the long run (Westerheide it explicitly distinguishes between short- 2006). term and long-term dynamics in the joint distribution of asset returns and inflation. In order to assess the inflation-hedging While error correction forms of the potential of the various asset classes vector-autoregressive model have been within a united framework, we follow extensively used in the macroeconomic the literature on predictability of asset literature in order to distinguish between returns (see Kandel and Stambaugh 1996; trends and business cycles and thus Campbell and Viceira 1999; Barberis 2000). between stationary and non-stationary As such, our analysis is closely related components in consumption and wealth to that of Hoevenaars et al. (2008), who dynamics (see Lettau and Ludvigson 2004 construct optimal mean-variance portfolios or Beaudry and Portier 2006 for recent with respect to inflation-driven liabilities studies), this approach is relatively new in based on model-implied forward-looking the finance literature. It has been used in variances and expected returns. Their basic modelling price and return dependencies finding is that alternative asset classes of financial securities (e.g. Blanco et al. add value to the investor's portfolio and 2005 or Durre and Giot 2007) and, more command signficant positive allocation recently, in inflation hedging contexts in the optimal mean-variance portfolio. (Westerheide 2006 or Hoesli et al. 2007). The authors further stress that the allocations in alternative asset classes are To the best of our knowledge, our paper higher within the optimal asset-liability is the first to provide a comprehensive portfolio as opposed to the optimal VECM model for the formal analysis of asset-only portfolio. In what follows, we inflation-hedging properties of various extend this existing literature in several traditional and alternative classes. We directions. First, we note that modelling derive econometric forecasts for model- 10 An EDHEC Risk and Asset Management Research Centre Publication
- 11. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Introduction implied volatilities and correlations, and overall allocation to the PSP while meeting we find that the results strongly deviate overall performance expectations, which from what is obtained with a standard VAR in turn allows better risk management model (see sections 3 and 4). Using the properties. VECM model that explicitly distinguishes between short-term and long-term The rest of this paper is organised as dynamics in the joint distribution of follows. In section 2, we present the asset returns and inflation, we identify econometric framework and motivate the presence of long-term cointegration the introduction on an error correction relationships between the return on typical extension of the standard VAR model. pension fund liabilities and the return of In section 3 we describe the database as various traditional and alternative asset well as preliminary statistical tests that classes. Our results suggest that real are used for model selection purposes. estate and commodities have particularly In section 4, we assess the inflation attractive inflation-hedging properties hedging potential of various traditional over long horizons. Subsequently, we use and alternative asset classes, and construct the VECM fitted parameters to perform a different versions of enhanced liability- simulation-based analysis of the impact hedging portfolios that contain real on ALM risk budgets of various portfolio estate and commodities in addition allocations. More precisely, the paper to TIPS. Section 5 analyses the impact suggests a structural form of the model in terms of risk budget improvements that incorporates i.i.d. innovations, which of the introduction of such enhanced allows for the generation of a stochastic liability hedging portfolios. Section 6 Monte Carlo analysis in a straight concludes. forward manner. The afore mentioned findings suggest that novel long-term liability-hedging investment solutions can be designed so as to decrease the cost of inflation insurance from the investor's perspective. In particular, it is possible to construct enhanced versions of inflation-hedging portfolios including inflation-linked securities, as well as commodities and real estate, so as to achieve satisfactory levels of inflation hedging over the long-term at a cost lower than that of versions that rely solely on inflation swaps. The intuition behind our results is rather straight forward. The increasedexpected return generated by making commodities and real estate in addition to TIPS part of the LHP allows a reduced An EDHEC Risk and Asset Management Research Centre Publication 11
- 12. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 Introduction 12 An EDHEC Risk and Asset Management Research Centre Publication
- 13. 2. Modelling Return and Inflation Dynamics A n E D H E C R i s k a n d A s s e t M a n a g e m e n t R e s e a rc h C e n tre Pub l i ca ti on 13
- 14. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 2. Modelling Return and Inflation Dynamics The first key challenge that needs to be while Δd-1 is not stationary (see Lütkepohl met for the analysis of the benefits of 1993). alternative investment strategies from an asset-liability perspective is the design One outstanding question, however, of an appropriate econometric model for is whether taking first differences and the joint distribution of asset returns and removing information related to price inflation. As recalled in the introduction, dependencies is too restrictive. The the bulk of the literature on stock return main concept behind the cointegration predictability and return dynamics framework, introduced by Engle and modelling (e.g., Campbell and Shiller 1988) Granger (1987), is precisely that integrated has relied on vector-autoregressive (VAR) variables may exhibit common trends that models. The calibration of a VAR model account for the non-stationary pattern in is generally performed on asset return the system. Thus, linear combinations of series corresponding to asset classes of non-stationary variables may turn out to interest and a set of potentially predictive be stationary. Formally, a process is said to economic variables that are introduced in be cointegrated of order (d, i) with d, i > 0, order to enhance the explanatory power of or yt ~ CI(d, i), if yt is integrated of order 3 - A process yt is stable if its reverse characteristic the model. In the context of VAR models, d (yt ~ I(d)) and there exists a linear polynomial has no roots in or the choice of using return series as opposed combination β'yt such that β'yt ~ I(d-i). on the unit circle: det (I - A1-…¡Apzp) = ∀|z|<1. to price series is either non-motivated, or In particular, if price series are I(1), we motivated by the stylised fact that return are interested in linear combinations of series are stationary while asset price these price series that are I(0) and thus series are not. The important consequence stationary. The next section introduces the of stationarity follows from the Wold's error correction version of the VAR model, Decomposition Theorem that states that known as VECM model, which incorporates stationary processes can be expressed as both levelled and difference series. a moving average process (cf. Hamilton In other words, the model uses 1994, p. 108), which in turn allows for systematic relationships of both the derivation of analytical expressions cross-sectional return dynamics and for shock responses, expected return and cross-sectional price dynamics. forward-looking variances. Secondly, non-stationarity leads to unstable, explosive and unbounded forward-looking variances 2.1 The econometric model and covariances, which makes the model Let us first consider the standard vector- non-tractable and non-suited for portfolio autoregressive (VAR) model of order p: selection purposes (Lütkepohl 1993). As a consequence, most econometric yt = c + A1yt-1 + … + Apyt-p + ut financial applications are based on return series or log-return series that are proven where yt represents a n x 1 vector of to be stationary, while price series tend endogenous variables, c is a constant, Ai to be integrated of order 1 at least. More are n x n coefficient matrices and ε is the formally, a process yt is integrated of order innovation process. If the process is stable,3 d (written: yt ~ I(d)) if Δdyt is stationary it may be rewritten as a finite moving 14 An EDHEC Risk and Asset Management Research Centre Publication
- 15. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 2. Modelling Return and Inflation Dynamics average process: lost, which could prove detrimental to the predictive power of the model. For (2) example, in an early economic study on the relationship between log consumption and log income, Davidson et al. (1978) have where Φ0 is the identity matrix and the Φs found that information on the deviation are recursively given as: from a long-term equilibrium enhances the explanatory power of the predictive model. To account for the presence of (3) long-term relationships in price series, the VAR model can be generalised through the with Aj =0 for all j > p. following error correction form (henceforth, VECM): Moreover, stability implies stationarity, meaning that expected returns, variances Δ yt = c + αzt-1 + Γ1Δyt-1 + … and covariances are time-invariant. This is + ΓpΔyt-p + ut (5) a critical point for financial applications where the econometric model is used for where z represents the deviation from the forecasting purposes and for deriving long term equilibrium. Subsequently, this analytical expressions for expected returns, representation has been used to define variances and correlations. In the case of the cointegration framework. Indeed, if a non-stationary processes, variances would long-term equilibrium relationship exists, be unbounded and time-variant, which it implies that z is a stationary variable. in turn leads to unbounded confidence Assuming that the long-term equilibrium intervals of the forecasted variables. can be expressed as a linear combination of the endogenous variables, we can rewrite In order to infer whether a time series is (5) as: stationary or not, various so-called unit root tests are available. The most commonly Δ yt = c + Πyt-1 + Γ1Δyt-1 + … used test is the Augmented Dickey-Fuller + ΓpΔyt-p + ut (6) (ADF) test (Greene 2003). If the test of unit roots cannot be rejected for all endogenous with the reduced rank matrix Π = αβ'. Its variables in y, the VAR process is integrated rank r<n determines the number of linear or cointegrated. As outlined above, if the independent long-term equilibrium variables are I(1), writing the VAR model (1) relationships and is also called cointegration on first-differenced variables generates a rank. In other words, there are r independent stationary process: linear combinations of the lagged endogenous variables that define the Δyt = c + Γ1Δyt-1 + … + ΓpΔyt-p + ut (4) cointegration relationships constituting r stationary variables β'y. Accordingly, The unfortunate consequence of using α and β are nxr matrices and β hosts the first-differenced variables is that any the cointegrating vectors so that β'yt. information regarding price dynamics is is stationary and reflects the long-term An EDHEC Risk and Asset Management Research Centre Publication 15
- 16. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 2. Modelling Return and Inflation Dynamics equilibrium relationships of the variables are driven by the structural responses to while α hosts the corresponding adjustment lagged innovations captured by Γi in (5). parameters, that is, the parameters that The cointegration relationship vector β determine the reversion speed to this and the reversion speed vector α govern long-term equilibrium. It is important to the long-run dynamics. More precisely, note that α and β are not unique due to the Π=αβ' induces instantaneous shocks reduced rank of Π. Accordingly, additional to the system if it deviates from the restrictions are needed to ensure that α and long-term equilibrium. As outlined earlier, β are just identified. A set of restrictions if the system is cointegrated, cross- that has been accepted as a standard sectional responses to shocks may be procedure in the econometric literature non-transitory or persistent as the is to define the upper part of β as the time-series "hang together". This section rxr identity matrix. Then, it can be shown introduces a modelling approach that that the estimated coefficient matrices are uses long-run and short-run restrictions asymptotically unbiased (Lütkepohl 1993, in order to identify the structural shocks p. 358). Two particular cases are worth of the system. The structural form of the mentioning. First, if r=0, then there are no model is characterised by i.i.d. innovations linear combinations of the original variables εt,i, as opposed to the correlated original that form a stationary process, and only innovation process ut,i. For this, we search first-differencing may lead to a stationary for the transformation matrix B such system. Secondly, if r = n, yt has a stable that: VAR(p) representation. ut = Bεt (7 ) Note that we specify all entries in y as the From the structural assumption we know log of the index level values. This is mainly that the covariance matrix of the i.i.d done because economic and financial time innovations ε (Σ ε) is diagonal. Without series are often shown to have exponential any loss of generality we postulate Σ ε trends. Writing linear regression models to be the identity matrix. Therefore, the such as the VAR or the VECM on the log original innovation covariance matrix may of these variables is therefore consistent be written as Σu = BB'. The transformation with the economically assumed dynamics matrix B hosts nxn parameters that need since taking the log of exponential process to be identified. Since Σ u is symmetric, linearises the processes. Log-returns are only n(n + 1)/2 independent equations are also convenient because they directly add available from Σu = BB'. For the parameters up across time-intervals. to be just identified we need n(n - 1)/2 additional restrictions. One way to do this is to use the Cholesky decomposition. 2.2 Structural model and Then, the contemporaneous impact matrix impulse-response functions B is given as a lower triangular matrix. In the context of cointegrated processes, As a matter of fact, the matrix B is not the dynamics of the underlying variables unique and the ordering of the variables may be separated in short-run and determines the dynamics of the structural long-run dynamics. Short-run dynamics shocks. 16 An EDHEC Risk and Asset Management Research Centre Publication
- 17. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 2. Modelling Return and Inflation Dynamics Given the specific context of the VECM and Rl =RΞB(In⊗Ξ). The matrix B can accordingly the fact that some shocks are persistent, be estimated by the maximum likelihood we follow Breitung et al. (2004) and impose method. Following Breitung et al. (2004), restrictions on short-run dynamics of the the corresponding log-likelihood function shocks as well as on the long-run is given by:4 impact matrix. According to Granger's representation theorem (Johansen 1995, theorem 4.2), this matrix is given by (see Vlaar 2004 for details): (11) Ξ = β⊥[α'⊥(I—Γ)β⊥ ]-1 α' ⊥B (8) Once the transformation matrix is identified, with α⊥ and β⊥ such that: we can write the structural VECM(1) (SVECM(1)) in its reduced form: α'⊥ α = β'⊥β (9) Δyt = c + Πyt-1 + ΓΔyt-1 + Bεt (12) The rank of this matrix is n-r since 4 - More details on distributional assumptions r cointegrating relationships form This representation allows us to conduct and asymptotic properties of stationary combinations, meaning that r structural impulse-response analyses, the estimation method can be found in Lütkepohl shocks have only transitory impacts and that is, to analyze the impact of isolated (2008). disappear in the long run. As can be seen independent structural shocks εt as opposed from (3), the long-run impact matrix of to the correlated innovation process ut in the structural shocks is ΞB. This matrix the reduced VECM form (5). To illustrate also has reduced rank equal to n-r. this, we write the model in its level variables Hence, at most r columns of ΞB can be form: zero columns since at most r structural innovation can have transitory effects and yt = c + A1yt-1 + A2yt-2 + Bεt (13) at least n-r shocks must have persistent effects to the system. Given its reduced with: rank, each column of zeros in ΞB accounts for n-r restrictions. As shown in Gonzalo and A1 = In + Γ + αβ' and A2 = —Γ (14) Ng (2001), the remaining restrictions split up into r(r - 1)/2 restrictions on the Accordingly, the VAR model structure allows transitory shocks (B) and into (n - r) us to write structural impulse-response ((n - r) - 1)/2 additional restrictions on functions as the permanent shocks (ΞB). We set the Ξ s = Φ sB (15) identifying restrictions through: with Φs as in (3). The elements in Ξs RΞBvec (ΞB) = rl and RBvec (B) = rs (10) illustrate the impacts of unit structural shocks that occurred at time t on the while we restrict rl and rs to vectors of endogenous variables at time t+s. The zeros. The long-run restrictions may kl-th element of Ξs, for instance, depicts also be written as Rlvec(B)=rl with the impact of a unit structural shock to An EDHEC Risk and Asset Management Research Centre Publication 17
- 18. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 2. Modelling Return and Inflation Dynamics variable l at time t on variable k at time t+s. As a result, this analysis allows us to measure the impact of inflation shocks on the various asset classes through time and to analyse short-run inflation-hedging potential, in addition to the long run dynamic analysis captured by the cointegration relations and the long-run dynamics given by Ξ∞. Confidence intervals for impulse- response functions are computed based on the bootstrap procedure described in Brüggemann (2006). In other words, we can assess co- or counter-movements between the inflation index and asset returns, and subsequently analyse horizon-dependent inflation hedging capacities for the various asset classes. It is also worth mentioning that the impulse response functions converge towards the long-run impact matrix ΞB. As a direct application of the structural VECM, section 5 uses the i.i.d. innovation process property to perform a Monte Carlo analysis based on the fitted model. The generated scenarios will be subsequently exploited in a portfolio construction context. 18 An EDHEC Risk and Asset Management Research Centre Publication
- 19. 3. Data and Model Specification A n E D H E C R i s k a n d A s s e t M a n a g e m e n t R e s e a rc h C e n tre Pub l i ca ti on 19
- 20. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 3. Data and Model Specification Our empirical analysis focuses on a set of are constant through time. Under this traditional and alternative asset classes. assumption, and further assuming that Stock returns are represented by the CRSP liability payments exhibit unconditional value-weighted stock index. Commodities inflation-indexation, the return on the are proxied by the S&P Goldman Sachs liability portfolio can be proxied as the Commodity Index (GSCI). Real estate return on a constant maturity zero-coupon investments are represented by the TIPS with a maturity equal to the duration FTSE NAREIT real estate index, which is of the liability cash-flows.7 We construct a value-weighted basket of REITs listed the time-series for such constant maturity on the NYSE, AMEX and NASDAQ. We zero-coupon TIPS in accordance with the thus limit the opportunity set to liquid methodology described in Kothari and and publicly traded assets. Finally, we add Shanken (2004), which states that the the Lehman Long US Treasury Index, as nominal return on a real bond is given well as the one-month Treasury bill rate.5 as the sum of a real yield plus realised Following the evidence from the extensive inflation. The real yield is in turn obtained literature on return predictability (see Stock as the difference of the nominal yield and and Watson 1999 among others), we also the sum of expected inflation plus the 5 - The series is downloadable from Kenneth add potential predictive economic inflation risk premium. As in Kothari and French's web site (borrowed variables to the set of endogenous Shanken (2004), we assume the inflation risk from Ibbotson Associates). 6 - See http://research. variables. We introduce the dividend premium to be equal to zero.8 We simplify stlouisfed.org/fred2. 7 - Incorporating additional yield (Campbell and Shiller 1988; Hodrick the computation of expected inflation by features such as actuarial 1992; Campbell and Viceira 2002), the taking it as the 60-month moving average uncertainty and inflation indexation would not credit spread (computed as the difference inflation. As a result, we obtain the returns impact the main message of the paper, which focuses between Moody's seasoned Baa corporate on liabilities as: on the inflation-hedging bond yield and the ten year Treasury t properties of real assets. 8 - Note, that Kothari constant maturity rate), as well as the rL,t = yield(τ) - Et(π) (16) and Shanken (2004) also considered the case of a term spread (obtained from the difference 50bps per annum inflation between the ten year Treasury constant where the upper index τ indicates the premium. 9 - Hoevenaars et al. (2008) maturity rate and the one month T-bill duration of the liabilities, which we have use seventeen years as the duration of the liability rate). The dividend yield data is obtained arbitrarily chosen in what follows to be portfolio. To avoid having from CRSP and all other economic figures equal to twenty years.9 In appendix A.2, to rely on interpolation, we rather use observable were obtained from the US Federal Reserve we provide a detailed derivation of the interest rate series from the FED, which are exclusively Economic Database.6 Our analysis is pricing scheme for inflation-indexed available for maturities based on quarterly returns from Q1 1973 liabilities. of one, two, three, five, six, seven, ten and twenty years through Q4 2007. (see http://research.stlouisfed. org/fred2). Regarding the liability side, we include 3.1 Model selection an inflation proxy represented by the We begin our empirical analysis with a consumer price index (CPI). As in Hoevenaars number of preliminary tests that help et al. (2008), we assume that the fund is us select the appropriate econometric in a stationary state as would be the case model. Table 1 presents the results of in a situation where the age cohorts and Augmented Dickey-Fuller (ADF) tests for the built-up pension rights per cohort level, differenced and twice differenced 20 An EDHEC Risk and Asset Management Research Centre Publication
- 21. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 3. Data and Model Specification series. The results clearly show that all price to move homogeneously with the remaining series (level series) are non-stationary and I(2) variables. Subtracting this variable from thus integrated of at least order 1. The the others transforms the variables from I(2) results further indicate that some economic into I(1) variables since the common trend variables are I(1), while other variables are is eliminated. A convenient choice for the I(2) as illustrated by the ADF tests on first and control variable in our setup is the price second differences. In fact, for all log asset index since subtracting it from the series return series as well as for the credit and means that the remaining I(2) variables will term spread series, we reject the hypothesis accordingly be transformed from nominal of a unit root. The corresponding price series to real variables. Additionally, the control are therefore integrated of order 1 and variable itself, in our case the price index, denoted by I(1)-variables. Other predictive needs to be replaced by its first difference. economic variables (dividend yield, ten-year This nominal-to-real transformation has yield, CPI) and the T-bill rate exhibit the been studied in Juselius (2007) and is inspired pattern of I(2) variables since taking second by preceding studies such as Engsted and differences of the original price series is Haldrup (1999), who consider various settings needed to eliminate the non-stationarity where some variables are I(2) and show that in the variables. adding their first differences as regressors to the model leads to consistent stationary The presence of I(2) integrated variables expressions for the long-run equilibrium. A is a concern since it implies that the drawback of this approach is that it supposes cointegrating relations may still exhibit price-homogeneity for the economic unit roots. As defined in Lütkepohl (1993), variables and thus a common trend in the the system is said to be integrated of order price level and the corresponding nominal 2, which we denote by yt ~ I(2), since the variables, a rather strong assumption that highest order of integration among the set needs to be empirically justified. A second of variables is 2. In order to address I(2) possibility to circumvent the problem of processes, the VECM model a priori needs I(2) variables is to find cointegrating to be extended to polynomial- or multi- relationships, that is, linear combinations cointegration frameworks, meaning that of I(1) and I(2) variables that are stationary. second differences, linear combinations Then, the system is said to be cointegrated of first differenced variables and linear of order (2; 0), yt ∼ CI(2; 0), meaning that combinations of level variables are included there exists a cointegration matrix β such in the econometric model (see Johansen that β'yt ∼ I(0). This is also referred to 1995 for more details on the methodology). as direct cointegration (Haldrup 1998). Two approaches have been proposed in the This approach is consistent with the econometric literature in order to circumvent recommendations in Hansen and the problem caused by such an increase Juselius (1995), who argue that ex-ante in the model complexity. A first approach cointegration rank tests should be consists of transforming the I(2) variable into accompanied with ex-post analysis of the I(1) variables without loss of information. cointegrating vectors. Accordingly, we will The idea is to choose a control variable first specify the cointegration rank r and among the I(2) variables that is supposed then try to find r linear combinations that An EDHEC Risk and Asset Management Research Centre Publication 21
- 22. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 3. Data and Model Specification form stationary variables. It is important to return dynamics depends merely on the note that, because of the normalisation in adjustment speed parameters displayed in the cointegration matrix β (see section 2.1), table 5. The next section presents model- different orderings of the variables lead to implied volatilities, correlations and impulse- different cointegrating relationships. response functions. In addition to the VECM estimates, we also estimate for comparison Table 2 presents the results of the Johansen purposes the standard VAR(1) model on trace test for the cointegration rank. The quarterly returns. Indeed, our goal is to show results suggest that the cointegration rank is that explicitly accounting for the presence either six or seven. We proceed as follows: for of cointegration relationships leads to each permutation of the order of variables, significantly different results from what we estimate the reduced form VECM with is obtained from the standard benchmark cointegration ranks r=6 and r=7 and extract VAR model used in previous literature. the time-series across the cointegrating vectors β'yt. Next, we perform ADF unit root tests in order to test for stationarity of the 3.2 Model implied variances, cointegrating vectors. The "best" ordering correlations and impulse is evaluated as the one that leads to the responses smallest p-values associated with the unit Based on the estimated structures, we derive root tests. Tables 3-5 yield the estimation for both the VECM and the VAR(1) model results for the "best" specification of the implied properties of the return dynamics. model measured by stationarity analyses In figure 1, we plot annualised volatilities on the cointegrating vectors. The p-values for returns on liabilities and asset classes associated with this order of variables range for different investment horizons according from 0 to 5.98% for the six vectors, which to the equations derived in appendix A.1. are reasonably low values. A particular focus of the graphs is on the difference between VAR-implied volatilities Table 4 presents information regarding (dashed lines) and VECM-implied volatilities the estimated cointegrating vectors. As (solid lines). The difference between the two evidenced by the results, all six equilibrium econometric methodologies turns out to relationships are quite similar with be rather significant for bonds, stocks and respect to the relevant, non-normalised commodities, with VECM-implied volatilities parameters. In fact, for each of the six proving to be significantly lower than first variables (liabilities, long bond, stocks, VAR-implied volatilities for these classes. CPI, Yield [10Y] and T-bills) the remaining Liability, T-bill and real estate returns show variables enter through a similar linear only minor differences between the VAR equilibrium relationship. While the loading and the VECM approaches. The difference on commodities and real estate is negative in implied volatility estimates is due to within the equilibrium relationship, credit the equilibrium reverting character of spread, term spread and dividend yield the additional part αβ'yt-1. As explained enter the long-term relationship positively. above, while β'yt-1 establishes the Accordingly, the interpretation of the impact equilibrium relationship, α determines the of the cointegrating relationship on the instantaneous impact of a deviation from 22 An EDHEC Risk and Asset Management Research Centre Publication
- 23. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 3. Data and Model Specification this equilibrium on Δyt. In an attempt to variables and negative relationships shed light on this discrepancy between for instantaneous covariances, as VAR- and VECM- implied characteristics for explained in Campbell and Viceira (2005). stocks and commodities, it is worthwhile For instance, stock returns exhibit some to refer to the credit spread variable. First, shown of predictability through lagged credit spread is highly significant as a dividend yields as evidenced by positive predictive variable for both stocks and coefficients in tables 3 and 9. At the same commodities. Secondly, credit spread enters time, contemporaneous innovations are positively into all cointegrating vectors with negatively correlated (see table 6). As a t-statistics ranging between eight and ten. result a positive shock in t on dividend Given that stock (respectively, commodity) yields goes (on average) along with a returns depend positively (respectively, negative shock on stock returns in t and, negatively) on changes in credit spread through the autoregressive link, a positive (see table 3) and given that (according to shock on stock returns in t+1. The offsetting table 5) the highest adjustment parameter effect may be interpreted as mean-reversion, values are negative (respectively, positive) which in turn lowers the volatility of the long-term dynamics partly offsets the compounded stock returns. However, our 10 - It should be noted that Campbell and Viceira (2005) short-term dynamics, which in turns reduces findings suggest that this predictability- have found with a VAR the volatility. For real estate and liabilities, induced mean-reversion effect is small model a steeper downward sloping volatility term this offsetting effect is less pronounced, compared to the mean-reversion effect structure for stock returns, but their analysis is based on as evidenced by the balanced set of induced by the long-term co-integration real return series, while we adjustment parameters in table 5, meaning relationships.10 explicitly analyse the nominal returns, and its inflation that some are positive while others are component. negative, which eventually cancels much Next, figure 2 displays horizon-dependent of the overall long-term impact on the correlation coefficients between liability corresponding return dynamics. returns and the return on various asset classes. The plots clearly suggest that A second remarkable effect is that bond, stock and real estate returns are VAR-implied volatilities seem to indicate negatively correlated with liabilities in that assets become more risky as the the short run, and that the correlation investment horizon increases, while VECM coefficient exhibits an upward pattern as volatilities have differing implications the investment horizon increases. Bond for the various assets. Liabilities, T-bills returns and stock returns start to become and real estate investments appear to be positively correlated with liability returns riskies in the long run, while bonds, stocks after about sixty quarters (fifteen years) and commodities exhibit a downward and end up with a significant positive sloping volatility structure, especially correlation of roughly 0.4 with a thirty-year from very short to medium-term investment horizon. Again, the result allows horizons. It should be noted at this stage us to identify significant discrepancies that a mean-reversion effect is already between VAR and VECM models, present in the VAR model, and is due to a especially for commodities and real estate. mean-reversion effect induced by the Commodity returns are positively predictive power of specific lagged correlated with liability returns in both An EDHEC Risk and Asset Management Research Centre Publication 23
- 24. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 3. Data and Model Specification cases, but the VECM implies a significantly higher and more stable correlation than the VAR. Model-implied correlations between real estate and liability returns, on the contrary, are significantly higher in the VAR model than in the VECM. This may be due to the fact that commodities are part of all six long-term equilibrium relationships, a result of the normalisation process of the matrix β and the variable order permutation procedure described above. The next section evaluates the impact of these model-implied moments and co-moments from a liability-hedging portfolio perspective. Impulse-response functions (figure 3) indicate that with the sole exception of commodities all responses to a structural liability shock are higher when implied by the VECM. This is mainly intrinsic to the model, as some shocks are persistent which cannot be the case in VAR model-implied shocks, an important restriction of the latter class of models. 24 An EDHEC Risk and Asset Management Research Centre Publication
- 25. 4. Inflation-Hedging Properties of Various Assets and Portfolios A n E D H E C R i s k a n d A s s e t M a n a g e m e n t R e s e a rc h C e n tre Pub l i ca ti on 25
- 26. Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management — January 2009 4. Inflation-Hedging Properties of Various Assets and Portfolios Allocation decisions dependent on the this intuition, we perform a scenario- investment zone have been widely studied based analysis to derive the funding ratio in the literature over the last decade. Brandt distribution at various investment horizons. (2005) and Campbell and Viceira (2005) The data generating process is described by discuss the differences between short- the vector error correction model (VECM) term or myopic and intertemporal asset introduced in section 2.1. We further use allocation decisions. The term structure the structural model so as to disentangle of risk, driven solely by the presence of the correlated innovation process and mean-reversion effects, with different transform it into i.i.d. innovations. We speeds of mean reversion (Lettau and draw i.i.d. random variables from the Wachter 2007), also plays a central role in multivariate standard normal distribution asset allocation decisions in the presence for the structural innovations εs (s = 1 … S) t of liabilities (see Campbell and Viceira 2005 and obtain the modelled returns by: for the notion of term structure of risk). This section uses VECM model-implied Δys t= c + Πyst-1+ Γyst-1+ Bεst (17) dynamics to assess inflation hedging potential across different investment for a total of S = 5,000 simulated paths. horizons. The first variable in yt represents the s liability return. We evaluate the different Consistent with the portfolio separation portfolios in terms of the funding ratio theorem, we will study the liability- (FR) distribution. The funding ratio at t in hedging portfolio (LHP) separately from scenarios s is accordingly given by: the performance seeking portfolio (PSP). s s In a framework where liabilities are indexed FRt = exp ((w' - ι)yt ) (18) with respect to inflation, and when short- term liability risk hedging is the sole focus, where ι denotes the n x 1 vector containing the optimal LHP allocation consists of a 1 in the first position and zeros investing 100% in the inflation-indexed elsewhere, and ω is the portfolio vector. bond portfolio (TIPS portfolio), which We first analyse the potential of unfortunately leads to very limited upside stand-alone inflation-hedging portfolios potential. Consequently, the investor needs before constructing optimal portfolios. a relatively sizable allocation to the PSP to meet the return requirements, which in turn generates a relatively high funding 4.1 Stand-alone hedging risk. Intuitively, one would expect that potential relaxing the constraint of a perfect liability This section assesses the inflation hedging fit for the LHP at the short-term horizon potential of the various asset classes on would allow one to include alternative a stand-alone basis. The analysis follows asset classes in the LHP, which in turn the methodology previously described, leads to increased upside potential. Overall, that is, all conclusions are drawn from the this would allow an investor to reduce funding ratio distributions over the 5,000 his/her allocation to the PSP, which leads simulated scenarios based on the fitted to a reduced surplus risk. To formalise VECM dynamics (cf. previous section). 26 An EDHEC Risk and Asset Management Research Centre Publication

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