Analysing and Decomposing the Sources of Added-Value of Corporate Bonds Within Institutional Investors Portfolios

Transcription

1 An EDHEC-Risk Institute Publication Analysing and Decomposing the Sources of Added-Value of Corporate Bonds Within Institutional Investors Portfolios August 2013 with the support of Institute

2 Table of Contents 1. Introduction The Investment Model Empirical Results...17 Conclusion...27 Appendix...31 References...39 About Rothschild & Cie...41 About EDHEC-Risk Institute...43 EDHEC-Risk Institute Publications and Position Papers ( ) This research is supported by Rothschild & Cie in the context of the The Case for Inflation-Linked Corporate Bonds: Issuers and Investors Perspectives research chair. Printed in France, August Copyright EDHEC 2013 The opinions expressed in this study are those of the author and do not necessarily reflect those of EDHEC Business School. The authors can be contacted at research@edhec-risk.com.

3 Foreword The present publication, Analysing and Decomposing the Sources of Added-Value of Corporate Bonds within Institutional Investors Portfolios, is drawn from the Rothschild & Cie research chair on The Case for Inflation-Linked Corporate Bonds: Issuers and Investors Perspectives at EDHEC-Risk Institute. The purpose of this research chair is to support research undertaken at EDHEC-Risk on the benefits of inflation-linked bonds from the investors perspective as well as from the issuers perspective. The chair also focuses on comparing and contrasting investors and issuers perceptions of inflation-linked bonds. The current paper, by Lionel Martellini, Scientific Director of EDHEC-Risk Institute, and Vincent Milhau, Deputy Scientific Director of EDHEC-Risk Institute, provides a formal analysis of the benefits of corporate bonds in investors portfolios, distinguishing between the impact of introducing them in performance-seeking portfolios and the impact of introducing them in liabilityhedging portfolios. The authors show that investor welfare can be improved by the design of performanceseeking portfolios with improved liabilityhedging properties, or conversely by the design of liability-hedging portfolios with improved performance properties. I would like to thank the co-authors, Lionel Martellini and Vincent Milhau, for their comprehensive analysis of the benefits of corporate bonds in investors portfolios. We would also like to extend our warm thanks to our partners at Rothschild & Cie for their collaboration on the project and their support of the research chair. Wishing you a pleasant and informative read, Noël Amenc Professor of Finance Director of EDHEC-Risk Institute An EDHEC-Risk Institute Publication 3

4 About the Authors 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, his work has been widely published in academic and practitioner journals and has co-authored textbooks on alternative investment strategies and fixed-income securities. Vincent Milhau is deputy scientific director of EDHEC-Risk Institute. He holds master's degrees in statistics (ENSAE) and financial mathematics (Université Paris VII), as well as a PhD in finance (Université de Nice-Sophia Antipolis). His research focus is on portfolio selection problems and continuous-time asset-pricing models. 4 An EDHEC-Risk Institute Publication

5 1. Introduction An EDHEC-Risk Institute Publication 5

6 1. Introduction 1 - On Friday October 19, 2012, it was reported that weekly inflows to investment-grade corporate bond funds were the highest in over two decades with $2.4 billion inflows into investment-grade corporate bond funds. Of that total, $1.4 billion flowed to investmentgrade corporate bond mutual funds, and the rest went into ETFs. 2 - While the presence of credit risk in corporate accounts is well understood and documented, sovereign credit risk is more difficult to apprehend given that sovereign state accounts are hardly audited. In this context, it can be argued that investment grade corporate bond markets offer better stability and visibility than sovereign bond markets. According to international accounting standards SFAS and IAS19.78, which recommend that pension obligations be valued on the basis of a discount rate equal to the market yield on AA bonds, the most straightforward way for pension funds to match liability payments is to build a portfolio of long-dated, investment grade corporate bonds. In practice, institutional investors including pension funds, but also insurance companies, sovereign funds, etc., are actually showing an increasing appetite for corporate bonds, not only for their liability hedging benefits, but also for their performance benefits related to the presence of a credit risk premium, which is imperfectly correlated with the equity risk premium. 1 This trend has been accelerated by the recent sovereign bond crisis, which has made high quality corporate bonds an attractive alternative, or a least complement, to Treasury bonds in investors portfolios. 2 This paper provides a formal analysis of the benefits of corporate bonds in investors portfolios, distinguishing between the impact of introducing them in performance-seeking portfolios and the impact of introducing them in liability-hedging portfolios. From a formal standpoint, our analysis is cast within the context of the liability-driven investing (LDI) paradigm, a disciplined investment framework that advocates splitting an investor s wealth between a dedicated liability-hedging portfolio (LHP) and a common performance-seeking portfolio (PSP), in addition to cash (Martellini (2006), Martellini and Milhau (2012)). While the LDI paradigm implies that investor welfare should depend on how good each building block is at delivering what it has been designed for (namely risk-adjusted performance benefits for the PSP and hedging benefits for LHP), the intuition suggests that the interaction between performance and hedging motives should also play an important role. We analyse this effect and show that investor welfare can be improved by the design of performance-seeking portfolios with improved liability-hedging properties, or conversely by the design of liability-hedging portfolios with improved performance properties. To see this, we first introduce a formal decomposition of investor welfare in terms of performance and hedging benefits, and show that a residual term remains, which can be interpreted as a cross-effect emanating from the interaction between performance and hedging motives. This result, which we call the fund interaction theorem, is important in that it shows that investor welfare indeed includes contributions from the PSP and the LHP, but also cross-contributions related to the hedging potential of the PSP. When negative, the cross-contribution signals the presence of a conflict between the performance and hedging motives, such as a short (long) position required for performance purposes and a long (short) position required for hedging purposes. This cross-contribution can be substantial for some parameter values, and sometimes equal or superior in magnitude to the performance and hedging contributions. The practical implications of the fund interaction theorem is that investors will in general benefit from improving hedging characteristics of the PSP, unless this improvement is associated with an exceedingly large decrease in Sharpe 6 An EDHEC-Risk Institute Publication

7 1. Introduction 3 - It is important at this stage to recognise that the fund interaction theorem and the fund separation theorem are not mutually inconsistent; in fact they co-exist within the framework of liabilitydriven investing, which the aforementioned results do not contradict, even though they do advocate a focus on the interaction between performance and hedging motives. ratio. 3 In the end, the net impact will be positive or negative depending on the relative strength of the following two competing effects. On the one hand, the PSP with improved hedging benefits can represent a higher fraction of the investor s portfolio for a given risk budget; on the other hand, the PSP with improved hedging properties may have a lower performance: hence the trade-off is between an increase in performance due to a higher allocation to risky assets, and a decrease in risk-adjusted performance due to a lower reward for each dollar invested. In an empirical analysis, we find that corporate bonds are particularly wellsuited to improve the PSP/LHP interaction, given that they have a well-controlled interest rate risk exposure while providing an access to an equity-like risk premium. In other words, they have on the one hand attractive interest rate hedging benefits which should help improve the correlation of the PSP with the liabilities compared for instance to an equity investment. On the other hand, they exhibit a higher expected performance compared to sovereign bonds due to the presence of a credit risk premium. These two properties make them natural candidates for inclusion in the performance portfolio, where the primary focus is on achieving a high expected return, and where a high correlation with liabilities helps to align performance and hedging motives, and also in the liability-hedging portfolio, where the primary focus is on interest rate risk hedging, and where the presence of a credit risk premium also contributes to aligning performance and hedging motives more effectively than what is allowed by sovereign bonds. As recalled above, if liabilities are discounted at the risk-free rate plus a spread, corporate bonds may actually hedge liability risk better than sovereign bonds do, precisely because they include a credit spread component that evolves in line with the discount rate on liabilities. The rest of the paper is organised as follows. Section 2 introduces a formal framework suited for an analysis of whether investor welfare can be enhanced by the introduction of corporate bonds in performance and/or hedging portfolios. Section 3 analyses the empirical implications of this framework, using monthly data over the period of February 1973-January Section 4 presents some conclusions and suggestions for further research. An EDHEC-Risk Institute Publication 7

8 1. Introduction 8 An EDHEC-Risk Institute Publication

9 2. The Investment Model An EDHEC-Risk Institute Publication 9

10 2. The Investment Model We consider a partial equilibrium model, with a single investor whose decisions have no impact on prices. Uncertainty in the economy is represented by a standard probability space, where is the set of measurable events and is a probability measure that represents the beliefs of the investor. Investment decisions are made at the initial date, denoted with 0, and the horizon is a positive date T. The investment universe consists of N risky assets whose returns over the period [0, T] are random as seen from date 0. We let denote the N 1 vector of realised returns, and and be the expectation and the covariance matrix of. In other words, is the vector of expected returns of the assets, while is their covariance matrix. The investor has an initial capital A 0, which he invests in the available assets, to end up with a terminal wealth A T. In addition, he is endowed liabilities, which are represented by a down payment L T at date T. 2.1 Portfolio Strategies and Investor Welfare The portfolio strategies that we consider in this paper are of the buy-and-hold type: that is, the investor chooses at date 0 the weights to allocate to the various assets, and then does not trade until date T. This assumption may seem at odds with the industry practice, which is closer to a fixed-mix strategy: that is, the portfolio has constant weights over time, as opposed to having constant numbers of shares of each asset. The reason why we focus on the buy-andhold case is because for such strategies, the moments of terminal wealth are simple functions of the weights chosen at date 0. The expectation and the variance of wealth at date T take the well-known forms: (2.1) It would also have been possible to cast the model in a fixed-mix framework, as Hoevenaars et al. (2008) do: assuming fixed-mix strategies and normal logreturns on securities, and using the loglinear approximation to the intertemporal budget constraint of Campbell and Viceira (2001), they are able to relate the moments of wealth to the constant weights. Our approach assumes a different form of strategies, which is perhaps less realistic, but it does not require any normality assumption about returns, and it gives exact, as opposed to approximated, expressions for the moments of wealth (see (2.1)). In other words, expressions (2.1) are model-free as far as the distribution of asset returns is concerned. In fact, because of the presence of liabilities, the investor is not so much concerned with the uncertainty in wealth itself as with the uncertainty in wealth relative to liabilities. Thus, the welfare measure has to be a function of some measure of the relative risk of the portfolio. Two main options have been proposed in the literature: a focus on the funding ratio, or a focus on the surplus A T L T. We follow the second approach here, because the assumptions made above imply simple expressions for the moments of the surplus. To write them, we define a few auxiliary notations: AL is the N 1 vector of covariances between the risky assets and liabilities, L is the expected 10 An EDHEC-Risk Institute Publication

11 2. The Investment Model 4 - The LHP that appears in the optimal strategy maximises the squared correlation, not the correlation itself, because this strategy is derived under the assumption that short-selling is permitted, so that a negative correlation can be virtually turned into a positive one. terminal value of liabilities, and variance of liabilities. Then: is the Finally, the welfare is represented by a quadratic utility from the surplus, a metric that depends on a risk aversion parameter γ: Pension funds often adopt liability-driven investing strategies, where performance and hedging are managed separately, through the allocation to a performance building block and a liability-hedging block. The objective of the performanceseeking portfolio (PSP) is to achieve a high Sharpe ratio, while that of the liabilityhedging portfolio (LHP) is to replicate the value of liabilities as accurately as possible. Thus, it is natural to restrict our attention to pairs of blocks that satisfy the following conditions: (2.2) where λ stands for the Sharpe ratio and ρ,l for the correlation with liabilities. The LDI portfolio is a combination of the two building blocks: maximum Sharpe ratio portfolio and the portfolio that maximises the squared correlation with liabilities. 4 The building blocks that we consider in this paper are not theoretical constructions that achieve the maximum Sharpe ratio or the highest squared correlation, and they only satisfy (2.2). In what follows, we denote with and the terminal wealths achieved by investing the capital A 0 in solely one building block. The following proposition gives expressions for the moments of the surplus under the strategy (2.3). This result is straightforward, as it is merely a rewriting of the expectation and the variance in the case where there are only two assets, namely a PSP and an LHP. Nevertheless, it is useful to justify the subsequent attempts to improve the performance of the LHP and the hedging properties of the PSP. Proposition 1 Consider strategy (2.3). The expectation and the variance of the surplus are given by: (2.4) (2.3) Such a strategy can be rationalised in an expected utility framework (see Martellini and Milhau (2012) for a continuoustime model), although the building blocks that would be involved in the utility-maximising portfolio rule are the For P = PSP,LHP, we have: (2.5) (2.6) An EDHEC-Risk Institute Publication 11

12 2. The Investment Model and the covariance term is: This proposition shows that although the objective of the LHP is to hedge liability risk and that of the PSP is to deliver a high risk-adjusted performance, the welfare is increasing in the Sharpe ratio of the LHP and in the correlation of the PSP with liabilities. Indeed, the partial derivatives of the quadratic utility with respect to ρ PSP,L and λ LHP are: These equalities have straightforward interpretations. The derivative of IW with respect to LHP is: (i) increasing in the allocation to the LHP (the properties of the LHP matter more when this block represents a substantial fraction of the global allocation); (ii) increasing in the volatility of the LHP (a large volatility tends to make the surplus more uncertain, which negatively impacts the utility, unless it is compensated by a high Sharpe ratio). Similarly, the derivative of IW with respect to ρ PSP,L is: (i) increasing in x; (ii) increasing in the risk aversion (highly risk-averse investors are more concerned with the variance of the surplus); (iii) increasing in the volatility of the PSP (the impact of a high volatility on the variance of the surplus needs to be compensated by a greater correlation with liabilities); (iv) increasing in the volatility of liabilities (hedging liability risk is more important when liabilities are particularly risky). 2.2 Comparing Strategies The quadratic utility serves as a criterion to compare different portfolio strategies: a strategy will be preferred to another if it yields higher utility. An interesting special case is that of two strategies that lead to the same surplus variance. Then, the difference between utilities reduces to the difference between expected terminal wealths: (2.7) where 1 and 2 refer to the two strategies. This situation is of particular interest because the difference between utilities does not depend on the risk aversion, which is hard to specify given that it is unobservable. Hence, a recurrent question in the empirical section of this paper will be to find a variance-matching allocation x 2 given a LDI strategy (PSP1, LHP1; x 1 ). Denoting with V i (x i ) the variance of the surplus under the i th strategy, we have to solve the equation V 2 (x 2 ) = V 1 (x 1 ) (note that the variance is indexed by i because it depends on the building blocks, not only on x i ). Proposition 1 shows that this equation is quadratic in x 2. As a consequence, it may have either one solution, two solutions or no solution at all (in the field of real numbers). The following proposition gives the conditions under which at least one solution exists, as well as the expressions for the solutions. Proposition 2 (Variance-Matching Allocations) Consider two LDI strategies characterised by the triplets (PSP1,LHP1; x 1 ) and (PSP2,LHP2; x 2 ). Let: 12 An EDHEC-Risk Institute Publication

13 2. The Investment Model 5 - In the absence of a riskfree asset, the Global Minimum Variance portfolio would replace the risk-free asset in the optimal strategy. If a > 0 and b 2 ac, then the allocations x 2 that satisfy V 2 (x 2 ) = V 1 (x 1 ) are given by: They are equal if b 2 = ac. Proof. See Appendix A. Note that these allocations do not necessarily lie between 0 and 1. Whether this condition is satisfied or not depends on other parameter values, and must be checked on a case-by-case basis. A special case where these allocations take a particularly simple form is when both strategies make use of a perfect LHP. Then, the surpluses for i = 1, 2 are zero, so that: One of these allocations is clearly negative, and is thus of limited practical interest. The other one is proportional to the ratio of the volatilities of the surpluses achieved by investing in PSP1 and PSP2, respectively. If PSP2 is a better hedge for liabilities, the surplus has lower volatility than the surplus, so that x 1 is greater than x 2. This result is natural: by improving the hedging properties of the PSP, one can allocate less to the LHP (that is, more to the PSP), without increasing the variance of the surplus under the LDI strategy. Because the PSP has generally higher expected return than the LHP, shifting the allocation towards the PSP allows the performance of the portfolio to be increased. In Section 3, we will quantify both the increase in allocation and the gain in expected return for realistic parameter values. 2.3 Fund Interaction Theorems The fund interaction theorems of Deguest et al. (2012) are derived in a continuoustime setting, with a Constant Relative Risk Aversion utility function. In this section, we derive similar results in our two-period setting. Even if they are not directly used in the empirical section of the paper (Section 3), they give another justification for the alignment of the performance and hedging motives. The first step is to derive the optimal strategy. For the simplicity of exposure, it is convenient to assume that there exists a risk-free asset, that is a security whose rate of return R ƒ over [0, T] is known as of date 0. This asset is thus a zero-coupon that redeems its principal at date T. We emphasise that it will not be used in the empirical analysis and is introduced here only to simplify the expression of the optimal strategy. 5 Mathematically, the optimisation program reads: (2.8) Proposition 3 (Optimal Strategy - Two-Fund Separation) The weights that solve Program (2.8) are: where the optimal PSP is the maximum Sharpe ratio portfolio and the optimal LHP is the portfolio that has the highest squared correlation with liabilities: λ PSP * and σ PSP* are the Sharpe ratio and the volatility of the PSP, and β L/LHP* An EDHEC-Risk Institute Publication 13

14 2. The Investment Model is the beta of liabilities with respect to the LHP. Proof. See Appendix B. The first block, denoted as PSP, is the maximum Sharpe ratio (MSR) portfolio computed over the investment universe. The optimal allocation to this fund is decreasing in the risk aversion. When γ grows to infinity, the optimal strategy is to allocate the weight β L / LHP* to the LHP, and the remainder to the risk-free asset. This ensures that the variance of the surplus is minimised. For a general risk aversion, the form of the optimal vector of weights is slightly different from that of the optimal policy in the continuous-time setting with CRRA preferences of Martellini and Milhau (2012). Indeed, the allocation to the LHP is independent from the risk aversion, while it is proportional to (1 1 / γ) in their paper. This dissimilarity arises because of the differences between the two models: in the one-period framework, portfolios cannot be rebalanced; and preferences are represented here by a quadratic utility, as opposed to a power utility function. It is worth noting that the optimal allocation to the PSP depends only on its Sharpe ratio and volatility, in addition to the risk aversion. The correlation between the PSP and liabilities does not impact the decision to invest more or less in this block. Similarly, the optimal portfolio rule does not depend on any performance indicator of the LHP: the only parameter of the LHP that matters is the beta of liabilities with respect to this portfolio. Overall, it may seem from Proposition 3 that the hedging properties of the PSP and the performance properties of the LHP are irrelevant to the investor. This view is, however, incomplete: while it is the case that these characteristics do not enter the optimal rule, it makes intuitive sense that they will eventually impact the welfare. For instance, if it happens that the investor holds a large long position in the PSP because this fund has a particularly nice Sharpe ratio, but the PSP is strongly negatively correlated with liabilities, this should increase the variance of the surplus, and, in turn, decrease expected utility. In the same way, if the LHP turns out to be a very good hedge of liabilities, with a beta close to 1, it will be desirable to invest a substantial fraction of wealth in it, but if it also has a very poor Sharpe ratio, this will tend to deteriorate the performance of the LDI strategy. The second fund interaction theorem (Theorem 2) gives a formal justification to these intuitions. Before we state it, we provide below a first interaction result, that gives a relationship between the parameters of the optimal PSP and those of the optimal LHP. It has the same form as the interaction equalities between the parameters of the PSP and those of the hedging portfolios derived in Deguest et al. (2012). Theorem 1 (1st Fund Interaction Theorem) The parameters of the optimal building blocks defined in Proposition 3 satisfy: Proof. See Appendix C. It should be noted that this equality holds only for the building blocks defined in Proposition 3, that explicitly maximise a criterion. It does not hold, in general, for the heuristic PSP and LHP that appear in (2.3). In words, this equality means 14 An EDHEC-Risk Institute Publication

15 2. The Investment Model that the relative domination of the PSP over the LHP in terms of Sharpe ratio (a domination that can be measured by the ratio of absolute Sharpe ratios, λ PSP* / λ LHP*, which is greater than 1 by definition of the two portfolios) is equal to the relative domination of the LHP in terms of correlation (which can be measured by the ratio ρ LHP*,L / ρ PSP*,L, also greater than 1). This result is useful to interpret the second fund interaction theorem, which we state now. Theorem 2 (2nd Fund Interaction Theorem) The maximum expected quadratic utility in (2.8) is: Proof. See Appendix C. Four contributions to welfare can be identified. The term is a contribution from the performance motive, since it depends only on the Sharpe ratio of the PSP. The term is a contribution from the hedging motive: the coefficient γ in this contribution reflects the fact that more risk averse investors attach more importance to hedging liability risk. The third term, ρ PSP*, L λ PSP* σ L is a cross-contribution of performance and hedging motives, since it involves one performance parameter (the PSP Sharpe ratio) and one hedging parameter (the correlation of the PSP with liabilities). By Theorem 1, it can equivalently be rewritten in terms of the LHP Sharpe ratio and the correlation of LHP with liabilities. Finally, the fourth contribution, equal to, does not depend on the strategy: it reflects investor s aversion for a high expected terminal value of liabilities (large μ L ) and a high uncertainty in this value (large σ L ). Among the first three terms, only the cross-contribution may be negative. The interpretation for its sign is as in Deguest et al. (2012). A positive crossterm means that the PSP is positively correlated with liabilities (at least to the extent that it has a positive Sharpe ratio). Hence, the long position in the PSP that is taken for performance considerations also contributes to reduce the surplus variance. The performance and the hedging motives are thus in line with each other, which results in a higher welfare. The magnitude of this increase in utility is itself increasing in the correlation of the PSP with liabilities. On the other hand, if the PSP covaries negatively with liabilities, the two motives compete with each other: the positive Sharpe ratio calls for a long position in the PSP, regardless of the correlation, but as a side effect, the negative correlation magnifies the surplus variance. This competition negatively impacts the welfare. An EDHEC-Risk Institute Publication 15

16 2. The Investment Model 16 An EDHEC-Risk Institute Publication

17 3. Empirical Results An EDHEC-Risk Institute Publication 17

18 3. Empirical Results The purpose of this section is to assess in a quantitative way the benefits of introducing corporate bonds in the PSP and/or in the LHP. The general idea is to compare portfolios that include these bonds to portfolios that exclude them, by reasoning with strategies that yield the same surplus variance. This is thus an empirical application of the methodology described in Section Dataset and Parameter Values In order to compute quadratic utilities and variance-matching strategies, we need the covariance matrix and the expected excess returns of the assets, and, as well as the vector of covariances between assets and liabilities. We calibrate these parameters to market data. Specifically, we consider an investment universe that contains one broad stock index, one sovereign bond index and one corporate bond index. The stock is represented by the CRSP valueweighted index of the S&P 500 universe, the sovereign index by the Barclays US Treasury bond index and the corporate index by the Barclays US Corporate Investment Grade index. All these indices are taken to be total return indices, that is, with dividends and coupons reinvested. The two bond indices are available from Datastream at the monthly frequency over the period of February 1973-January 2013, and the stock index is taken from CRSP. Figure 1 shows the values of $1 invested in each index on February 28, 1973, and held until December 31, Table 1 reports descriptive statistics on the three series, obtained from simple monthly returns. The stock turns out to have the highest volatility, of 17.2% per year. The least volatile asset is the nominal bond, with a standard deviation of 5.7% per year, and the corporate bond lies between the other two assets. The correlation between the two bond indices is as high as 82.1%, which suggests the presence of a set of common factors in their returns. The sample mean returns (not shown in the table) are respectively 11.27% per year for the stock, 8.08% for the sovereign bond and 8.63% for the corporate bond. These values should be taken cautiously, because it is well-known (see e.g. Merton (1980)) that sample means are poor estimators of expected returns. In particular, the expected return on sovereign bonds seems overstated: as pointed by Dimson et al. (2008), US bonds performed well between 1981 and 2000, with average nominal return close to 12% per year, but the period from 1946 to 1981 saw a bear market, with nominal returns around 2% per year and negative real returns due to the high inflation rates recorded in those years. Hence, a value of 7% for Treasuries seems more in line with the examination of long-term series. It still appears to be an optimistic forecast in view of the low current level of interest rates: indeed, the presence of mean reversion in rates makes it more likely for bond prices to go downwards than upwards in the future. It is expected that corporate bonds earn a higher return than sovereigns, due to the default risk premium that they embed. But precisely because they are subject to the default risk of the issuer, the higher return that they promise can be offset by the loss of value in case of default or credit events such as downgrading. As a matter of fact, Dimson et al. (2008) find that the differential annualised return between US nominal and corporate bonds is about 18 An EDHEC-Risk Institute Publication

19 3. Empirical Results 6 - A broad stock index such as the S&P 500 has poor inflation-hedging properties, but some sectors such as Oil & Gas and Technology have higher betas with inflation (Ang et al., 2012). 50 basis points per year between 1900 and 2000, while the difference in yields, which measures the premium that investors require to bear the default and credit risks, is about twice as high. The 50 basis points value roughly corresponds to the difference [8.63% 8.08%] in our dataset. Nevertheless, the average duration of the corporate index is 6.7 years, while that of the Treasury index is 4.7 years. Hence, the expected return on the former should include a maturity premium in addition to the default premium. We account for this by assuming an expected return of 8.4% for corporate bonds. Finally, equities are the most risky asset class of the three, and as a consequence, should have the highest expected return. This effect is already present in the data, since the mean return is 11.27%. But Dimson et al. (2008) find a premium of 5% over long bonds. Since we have assumed that nominal bonds have an expected return of 7% per year, we thus round the sample value to 12%. To compute the Sharpe ratios of the asset classes, we take the risk-free rate to be the average over the period of the secondary market rate on 3-month Treasury bills, a series that we obtain from the website of the Federal Reserve of St. Louis. This average rate is 5.29%, which implies the Sharpe ratios 0.39 for stocks, 0.30 for sovereigns and 0.40 for corporates Building Blocks The LDI strategy requires a PSP and an LHP. In what follows, we consider three different versions of the PSP with an increasing fraction of wealth allocated to corporate bonds: the benchmark PSP, also referred to as PSP 1, contains 80% of equities, the remainder being invested in Treasuries. The other two PSPs respectively contain 15% and 30% of corporates. In order to fix the degree of freedom, we impose that the ratio of stock weight to sovereign weight be the same in all three PSPs. This leads to: These PSPs are all dominated by equities, since it is the asset class that is typically used to seek performance. But none of them is meant to be an MSR portfolio. In the same way, we consider three different versions of the LHP: the benchmark LHP, also referred to as LHP 1, is invested in sovereign bonds only, and the other two LHPs contain respectively 25% and 100% of corporate bonds. 3.2 Introducing Corporate Bonds in PSP or LHP To illustrate the benefits of introducing corporate bonds in the portfolio, we assume that the investor follows a LDI strategy of the form (2.3). The investment universe consists of the stock, the sovereign bond and the corporate bond. All these portfolios are invested in bonds only. Since LHPs are most often dominated An EDHEC-Risk Institute Publication 19

20 3. Empirical Results 6 - A broad stock index such as the S&P 500 has poor inflation-hedging properties, but some sectors such as Oil & Gas and Technology have higher betas with inflation (Ang et al., 2012). 7 - Of course, nominal bonds are not good hedges for inflation-indexed liabilities with a short duration, because realised inflation risk becomes the dominant source of risk when the maturity shortens. 8 - Campbell et al. (2009) show that the 10-year breakeven rate in the US was relatively stable between 2004 and 2008, but collapsed at the end of 2008, which highlights the fact that nominal and real rates are not perfectly correlated. by nominal bonds, the LHP 1 is an acceptable stylised representation of the typical LHP held by a pension fund. In general, the optimal LHP (LHP in Proposition 3) may contain stocks, depending on the risk factors to hedge. For instance, if stocks are good at hedging inflation risk, then they can be useful for hedging inflation-linked liabilities. 6 In our model, liability payments are not indexed on inflation, so the value of liabilities is only impacted by the discount rate. It is natural to seek to hedge this risk with nominal bonds. That is not to say that stocks do not have any interest-rate hedging properties, but their duration is more difficult to evaluate than that of bonds. For instance, Leibowitz et al. (1989) argue that the traditional Dividend Discount Model overstates the duration of equities, and Reilly et al. (2007) show that the empirical duration of the S&P 500 exhibits large fluctuations over time. In contrast, a bond index, which has a built-in exposure to interest rate risk, has a much less volatile duration. Were liabilities indexed on inflation, one could also use the obvious option to introduce inflation-linked bonds in the LHP. Nevertheless, nominal bonds are in general dominant in LHPs. This predominance can be explained by practical reasons: the supply of nominal bonds still exceeds that of inflation-linked bonds, although indexed debt represents a growing fraction of the outstanding debt of large developed countries, so these bonds are more liquid. One can also give theoretical arguments that justify to some extent the use of nominal bonds in the LHP. Indeed, interest rate risk is the dominant source of risk in liabilities, because the sensitivity of liabilities to the discount rate is increasing in their duration, which is typically long. Even for inflation-indexed liabilities, interest rate risk dwarfs realised inflation risk, at least when liabilities have a long maturity (Martellini and Milhau, 2013). 7 As argued by Campbell et al. (2009), nominal bonds are in fact perfect substitutes for inflation-linked bonds if breakeveninflation rates, defined as the differences between nominal rates and real rates of the same maturity, are constant. 8 Table 2 gives descriptive statistics for the six building blocks. For comparison purposes, it also provides the same statistics for the stock (the figures for the sovereign bond and for the corporate bond are read in columns LHP 1 and LHP 3, respectively). All these figures have been obtained from the descriptive statistics for the asset classes shown in Table 1. In detail, we compute the expected excess return on a portfolio with weights as, so the annualised expected excess return can be derived from those of the constituents as: where T is the investment horizon. The variance of the portfolio is, where is the covariance matrix of the assets: its coefficients are given by σ ij = σ i σ j ρ ij, where ρ ij is the correlation between the assets and σ i is the volatility of asset i at horizon T, which can be computed from the annualised volatility and expected return as: 20 An EDHEC-Risk Institute Publication

21 3. Empirical Results 9 - We note that the portfolios respect the condition on Sharpe ratios and correlations with liabilities (see (2.2)). The annualised volatility of the portfolio is then computed as: For portfolios invested in a single asset, these definitions lead to and, as expected, but for portfolios invested in multiple asset classes, and depend on the horizon T. We set T to 10 years. Unsurprisingly, introducing corporates in the LHP has a positive effect on expected returns, which is a direct consequence of the outperformance of these bonds over their sovereign counterparts. It also increases the volatility, but given the assumed risk-free interest rate (5.28%, which is the sample average of the 3-month T-bill rate), it has also a positive effect on the Sharpe ratio. For the PSP, however, including corporates comes at the cost of a lower allocation to stocks. Because stocks have a higher expected return, the PSPs that include corporates have lower expected returns than the PSP invested only in stocks and sovereign bonds (PSP1). We assess the liability-hedging properties of the various portfolios with respect to two liability processes. First, we consider liabilities discounted at the risk-free rate: they are proxied by the sovereign bond index itself. The second liability value is obtained by adding a spread to the risk-free rate. This is motivated by the international accounting standards SFAS and IAS19.78, which recommend that pension obligations be valued on the basis of a discount rate equal to the market yield on AA corporate bonds. For simplicity, we refer to this rate as the risky rate in what follows. It should be noted, however, that the risk-free rate is also risky in the sense that it varies over time. Thus, the name risky simply implies that the AA rate proxies the discount rate that applies to a defaultable corporate bond rated as investment grade. We represent this second liability process by the value of the corporate bond index. In addition to the correlations between the various portfolios and the two liability values, we also report a related indicator, which is the volatility of the final surplus, computed as the square root of: where ρ PL is the correlation between the portfolio and liabilities. This variance depends on the initial wealth, A 0, and the initial value for liabilities, L 0. We assume that both these values are equal (initial funding ratio of 1), and we normalise them to 1. Correlations and the surplus volatilities are also reported in Table 2. 9 A first observation is that all three PSPs are more correlated with liabilities discounted at the risky rate than with those discounted at the risk-free rate. This is due to the dominance of stocks and corporate bonds over Treasuries in the PSP: by definition of the liability process, corporates are a perfect hedge for liabilities discounted at the risky rate, and as shown by Table 1, stocks have a correlation of 35.6% with this process, versus 8.50% only with liabilities discounted at the risk-free rate. In spite of this higher correlation, the surpluses computed with respect to liabilities discounted at the risky rate are more volatile, which comes from the higher volatility of these liabilities. It also appears that introducing corporates in the PSP leads to higher correlations with both liability An EDHEC-Risk Institute Publication 21

22 3. Empirical Results processes: for instance, when the risky rate is used, the benchmark PSP (PSP 1) has a correlation of 39.7% with liabilities, while the PSP that contains 30% of corporates (PSP 3) has a correlation of 52.9%. This comes as no surprise, since corporates are by definition a perfect hedge for liabilities, but we also see that the correlation increases when the risk-free rate is chosen: indeed, PSP 1 is largely dominated by stocks, that have little hedging capacity with respect to nominal interest rate risk, so adding bonds can only enhance its hedging properties. But this improvement is paid by a decrease in expected return, from 11.2% for PSP 1 to 10.4% for PSP 3. This is a consequence of the decrease in the allocation to equities, which are the best-performing asset class in our model. Overall, introducing corporate bonds in the PSP gives rise to a tradeoff between performance (measured by expected return) and correlation. The situation is different for the LHP. The benchmark LHP is entirely invested in the asset class with the lowest expected return (7.0%), so that increasing the allocation to corporate bonds gives access to higher performance. When liabilities are discounted at the risk-free rate, this even generates a higher correlation with liabilities: indeed, the correlation between these liabilities and the LHP invested only in corporate bonds is perfect, by construction. Hence, in that case, there is no trade-off between performance and hedging: using corporate bonds leads to an improvement on both fronts Benefits of Introducing Corporate Bonds in PSP As argued previously, the introduction of corporate bonds in the PSP improves its correlation with liabilities, but lowers its performance compared to an equitydominated portfolio. It remains to assess the net impact of such a modification in the context of an ALM strategy. The intuition is as explained in Section 2.2: by choosing a portfolio that has better correlation with liabilities, one can decrease the allocation to the LHP, without increasing the risk of the global portfolio. In other words, one can afford investing more in the PSP with improved hedging properties, while staying within a fixed risk budget, which is equal to the volatility of the LDI strategy that uses the benchmark portfolio. Even if this higher weight is applied to a portfolio that has lower expected return (see Table 2), it is expected that the final effect on the expected return of the strategy will be positive. In order to verify whether this intuition is true, we consider a benchmark LDI strategy, invested in the benchmark PSP (denoted with PSP 1 in Table 2), and in the LHP that only contains Treasuries (denoted with LHP 1 in Table 2), and an improved LDI strategy, invested in the same LHP and an improved PSP, either PSP 2 or PSP 3. For each allocation x 1 to the benchmark PSP, we seek the variance-matching allocations to the improved PSPs that contain corporate bonds. We then compute the difference between the expected terminal wealths attained with the benchmark LDI strategy and with the improved LDI strategies. Since we have normalised the initial asset value to $1, this difference can be interpreted as a difference between expected arithmetic returns. But because returns are usually expressed in annual terms, we instead report the difference between annualised arithmetic returns, that is, if is the wealth attained at date T with the benchmark LDI strategy and is the 22 An EDHEC-Risk Institute Publication

23 3. Empirical Results wealth attained with the improved one (i = 2 or 3, depending on the PSP used): Note that this quantity is different from the change in quadratic utility (see (2.7)), which is, but it is more interpretable, since it represents a gain (or a loss, depending on the sign) in annual expected return. Besides, it has the same sign as IW i IW 1. As pointed in Section 2.2, there are in fact two variance-matching allocations x i for each value of x 1. Let us denote with and these two values, which are defined in Proposition 2. Since distinct values of xi lead to different values for, we have to decide which one to pick. A meaningful criterion is the sign of x i and that of 1 x i : for the LDI strategy (2.3) to involve long positions in the PSP and the LHP, x i must be between 0 and 1. Thus, a natural criterion would be to retain the root of the variance-matching equation that lies within these bounds. This raises the question whether there exists exactly one nonnegative solution. This is the case if the conditions of Proposition 2 are met, and the quantity is negative. The sign of c depends in general on parameter values, but it is likely to be positive: indeed, the variance of the surplus achieved by investing only in LHP 2 will be in general small, at least smaller than the variance achieved with the LDI strategy. Thus, one can reasonably expect to have exactly one positive solution, which is the higher of the two roots: thus, we retain as the variance-matching allocation to the improved PSP. With our parameter values, it turns out that is always the unique positive solution. Nevertheless, it can be larger than 1, which would yield short positions in the LHP. Thus, we only report the value of for those values of x 1 that imply a long position in the LHP. Figure 2 shows the variance-matching allocation and the increase in expected return, for x1 ranging from 10% to 80%. It appears that is larger than x 1 for all tested parameter values, which justifies the aforementioned intuition: for a given risk budget, one can invest more in a PSP with improved liability-hedging ability. The increase in weight is also larger with the PSP that contains 30% of corporates (PSP 3) than with the one that contains only 15% of these bonds (PSP 2). For instance, using the risk-free rate plus the spread to discount liabilities, and assuming an allocation x 1 = 40% to the benchmark PSP, one can invest 47.5% in PSP 2, and 58.0% in PSP 3. Lower values would be obtained with liabilities discounted at the risk-free rate: respectively, 46.2% in PSP 2, and 54.4% in PSP 3. This makes sense: by introducing corporate bonds in the PSP, one improves primarily its correlation with the liability process whose discount rate includes a credit spread component, so the increase in the PSP weight should be higher than for liabilities discounted without this spread. A related observation is that the increase in annual expected return is larger for the former liability process: still focusing on the case x 1 = 40%, one achieves modest increases of 5.5 and 12.6 basis points per year respectively, by investing in PSP 2 and PSP 3 under the valuation rule at the risk-free rate, and these values grow to 10.5 and 24.6 annual basis points if the spread is added. One may be puzzled by the observation An EDHEC-Risk Institute Publication 23

24 3. Empirical Results that the gains in performance do not converge to zero as x1 approaches zero. Indeed, if one does not invest in the PSP, then one should be indifferent to a change in its composition. But this statement is incomplete: allocating nothing to the benchmark PSP does not imply that one does not invest at all in the improved PSP. To see this, let us consider the variancematching condition written for the same pair of building blocks: it reads V1(xi) = V1(x 1 ), and is of course satisfied for x i = x 1. But because it is a quadratic equation in xi, it has in general two roots, and. For x 1 = 0, the expressions of Proposition 2 imply that: so is zero if, and only if, is nonnegative. Given our parameter values, b is actually negative, so x+i is positive (the zero root is, which is the dropped solution). This explains why it is possible to enjoy an increase in expected return even if one invests nothing in the benchmark PSP. In order to provide a visual representation of the changes in the properties of the strategies, Figure 3 shows the distributions of the surplus under the three LDI strategies that are being compared, for an allocation x 1 = 50% to the benchmark PSP: the two improved strategies are invested in the same LHP, and in PSP 2 and PSP 3 respectively, and they satisfy the variance-matching conditions V 2 (x 2 ) = V 1 (x 1 ) and V 3 (x 3 ) = V 1 (x 1 ). Because our theoretical model does not assume a particular distribution for asset returns, we need to specify one at this stage in order to simulate values for the surplus. We postulate a log-normal distribution for returns, and we estimate the moments (vector of expected returns and covariance matrix)of this distribution as the sample moments of logarithmic returns of the stock and the two bond indices. Thus, these moments are slightly different from those given in Table 1. We thensimulate 50,000 values for the surplus attained with each LDI strategy after ten years. By construction, all three strategies imply the same surplus volatility (57.5%), but the expected surpluses are higher with the strategies that allow for corporate bonds (20.7% and 24.1% of A 0 respectively) than with the benchmark strategy (18.2% of A 0 ). The simulated distribution also enables to estimate the probability of a negative surplus after ten years: it appears that this probability is decreasing in the weight of the corporate bonds in the PSP Benefits of Introducing Corporate Bonds in LHP We now conduct a similar but dual analysis in order to measure the gains of introducing corporate bonds in the LHP. The primary objective pursued by adding corporate bonds to nominal bonds in the LHP is not an improvement in correlation with liabilities (as for the PSP), but a performance gain. This, of course, does not rule out the possibility of enjoying a better correlation: Table 2 shows that with the valuation rule at the risky rate, the LHPs that contain corporates have both greater expected return and larger correlation. Thus, the mechanism at work here is different than the one in Section 3.2.2: by increasing the performance potential of the LHP, one can allocate more to the LHP without lowering the expected return of the LDI strategy. This higher weight in 24 An EDHEC-Risk Institute Publication

25 3. Empirical Results the LHP should lead to a lower risk for the strategy. If liabilities are discounted at the risk-free rate, a trade-off between the two effects arises, because the LHP with corporates has lower correlation than the benchmark LHP invested in sovereign bonds only. On the other hand, if the discount rate includes a spread, there is no tradeoff since introducing corporates in the LHP also enables its correlation to increase. In order to have a quantitative measure of the reduction in risk, we take the following approach: for a given allocation y 1 = 1 x 1 to the benchmark LHP, we search for the allocation y i = 1 x i to the improved LHP, which can be either LHP2 = (0, 0.75, 0.25) or LHP3 = (0, 0, 1), that leaves the expected terminal wealth unchanged. Denoting with E i (x i ) the expected terminal surplus in Proposition 1, with i = 2 or 3 depending on the LHP used, we can write this expectation-matching condition as E i (x i ) = E 1 (x 1 ). This equation is linear in x i, so it has at most one solution, which is given by: (there may be no solution in the very special case where, but this equality is unlikely to hold true, because the PSP has in general higher expected return than the LHP; at least Table 2 shows that this condition is verified with our parameter values). We then compute the relative change in the volatility of the final surplus, that is the ratio: where is the final wealth achieved with the LDI strategy invested in PSP 1 and LHP 1 and is the wealth attained with the LDI strategy invested in PSP 1 and LHP 2 or LHP 3. Results are shown in Figure 4. First, it appears that the use of corporate bonds in the LHP allows the allocation to this building block to be increased without adversely impacting the expected return of the strategy. This follows from the outperformance of corporate bonds with respect to sovereign bonds. Second, this modification of the composition also leads to a lower surplus volatility. Since the increase in the LHP weight is more important for the LHP invested in corporate bonds only, the reduction is risk is more substantial with this LHP. This reduction is also larger when liabilities are discounted at the risk-free rate plus spread, which is natural given that corporate bonds are a perfect hedge for this liability process. Overall, the relative decrease in standard deviation can be very significant. For instance, focusing on the case where liabilities are discounted at the risky rate and an allocation 60% to the benchmark LHP, one decreases the volatility by 12.6% by investing in LHP 2, and by 64% if one chooses the LHP fully invested in corporate bonds. Following the same protocol as in Section 3.2.2, we then simulate the distributions of surpluses achieved with three LDI strategies, invested respectively in (PSP 1, LHP 1), (PSP 1, LHP 2) and (PSP 1, LHP 3). The weight allocated to LHP 1 in the benchmark strategy is 50%, and the allocations to LHP 2 and LHP 3 are computed in such a way as to keep the expected terminal wealth unchanged, or equivalently, the expected An EDHEC-Risk Institute Publication 25

26 3. Empirical Results terminal surplus fixed (at 18.2% of A 0 ). Thus, what is gained by introducing corporate bonds is a reduction in the volatility of the surplus, from 57.5% of A 0 with the benchmark strategy, to only 33.6% with the strategy that uses corporate bonds as the unique hedging instrument for liability risk. The probability of underfunding is also reduced. 26 An EDHEC-Risk Institute Publication

27 Conclusion An EDHEC-Risk Institute Publication 27

28 Conclusion This paper provides an empirical analysis of the sources of added-value of corporate bonds for institutional investors. For reasonable parameter values, we have found corporate bonds to be attractive additions to investors portfolios. On the one hand, introducing them in performance-seeking portfolios (PSPs) typically generates positive benefits from an asset-liability management perspective since it will lead to substantial improvements in hedging benefits, which come at the cost of a less-thanproportional reduction in performance compared to equity-dominated portfolios. Introducing corporate bonds in liabilityhedging portfolios (LHPs) is also found to generate a positive impact on investor welfare since it leads to improvements in both hedging and performance benefits, especially for investors facing liabilities discounted using a credit spread adjustment. In this context, one may want to assess whether a particular proportion of corporate bonds in each portfolio would lead to the highest level of welfare gains. The answer to this question is of course that the optimal allocation to corporate bonds should maximise the Sharpe ratio within the PSP, and the (squared) correlation with the liabilities within the LHP. In fact, the fund interaction theorem and the fund separation theorem are not mutually inconsistent; the fund interaction theorem complements the fund separation theorem by emphasising the benefits of having, if and when possible, an alignment, as opposed to a conflict, between the performance and hedging motives. Our analysis can be extended in many possible directions. First of all, it would be useful to draw a distinction between investment grade bonds and high yield bonds, given that the risk, return as well as factor exposure characteristics of these two segments of the corporate bond market are clearly distinct. Also, one could consider corporate bond benchmarks with various numbers of constituents so as to assess the impact of credit risk diversification on the results in the paper. Finally, the interaction analysis in this paper could usefully be extended to other types of securities, including, for example, equities. In particular, a large crosssectional variation exists in the interest rate exposure of various segments of the equity markets, and some sectors (e.g., utilities), or types of stocks (e.g., high dividend stocks), have been found to have a duration that is substantially higher than the duration of a well-diversified equity portfolio (see for example Reilly et al. (2007)). If an equity benchmark can be constructed with a substantially enhanced interest rate risk exposure, without a significant cost in terms of Sharpe ratio, then the analysis in this paper suggests that using this benchmark as opposed to a standard market index may lead to improvement in welfare for an investor facing liabilities. More generally, there are at least two reasons why investors would find it optimal to invest in any asset class. First, the asset class under consideration can be useful if it provides access to excess performance with respect to cash (speculative demand for risky assets that generate an exposure to rewarded sources of risk). Secondly, it can also be useful if it provides hedging benefits (intertemporal hedging demands for risky assets that 28 An EDHEC-Risk Institute Publication

29 Conclusion can be used to immunise wealth levels with respect to risk factors that impact the value of liabilities and the opportunity set). Clearly, these motives are not mutually exclusive; for example, bonds are useful ingredients in the speculative component of investors portfolios, where they bring excess performance with respect to cash and diversification benefits with respect to equities, and they are also useful in the hedging component of liability-driven long-term investors portfolios, where they allow for protection against unexpected changes in interest rates. The analysis in our paper suggests that not only are the two motives not mutually exclusive, but also that these two motives interact, and their interaction is a key driver of the ability for investors to achieve attractive funding objectives from an asset-liability management perspective. An EDHEC-Risk Institute Publication 29

30 Conclusion 30 An EDHEC-Risk Institute Publication

31 Appendix An EDHEC-Risk Institute Publication 31

32 Appendix Figure 1: Time series of bond indices. This figure shows the values of $1 invested in the S&P 500 Composite, the Barclays US Treasury bond index and the Barclays US Corporate Investment Grade index respectively. Table 1: Descriptive statistics on state variables. In panel (a), off-diagonal elements are correlations, and diagonal elements are annualised standard deviations. These statistics are based on arithmetic monthly returns over the period February January In panel (b), expected returns are set to fixed values. Table 2: Statistics for building blocks (in %). This table contains the annualised (arithmetic) expected returns, volatilities and Sharpe ratios of the stock (S) and the six building blocks, as well as their correlations with liabilities and the variances of the terminal surpluses, assuming an initial funding ratio of 1. Liabilities are either discounted at the risk-free rate ( Rf ) or at the risk-free rate plus the AA spread ( Rf+sp. ). The risk-free rate used to compute the Sharpe ratios is the average 3-month rate over the period considered (5.28%). The compositions are as follows (stocks, sovereign bonds, corporate bonds): PSP1 = (0.80, 0.20, 0); PSP2 = (0.68, 0.17, 0.15); PSP3 = (0.56, 0.14, 0.30); LHP1 = (0, 1, 0); LHP2 = (0, 0.75, 0.25); LHP3 = (0, 0, 1). 32 An EDHEC-Risk Institute Publication

33 Appendix Figure 2: Introduction of corporate bonds in the PSP Increase in annual expected return and in PSP allocation. Each panel in this figure compares two LDI strategies. The first one ( benchmark ) is invested in the benchmark PSP PSP1 = (0.80, 0.20, 0), and in the LHP LHP1 = (0, 1, 0) (stocks, sovereign bonds, corporate bonds). The second one ( improved ) is invested in an improved PSP, PSP2 = (0.68, 0.17, 0.15) or PSP3 = (0.56, 0.14, 0.30), and in the same LHP as the benchmark one. For each allocation to the benchmark PSP, the weight to invest in the improved PSP is chosen so as to keep the variance of the surplus unchanged. The outperformance of the improved LDI strategy is measured as the gain in annual expected return after ten years, expressed in basis points. The circled reference lines relate to the benchmark strategy invested in PSP 1 and LHP 1. Panels (a) and (b) differ through the liability process: liabilities are discounted at the risk-free rate in Panel (a), and at the risk-free rate plus the spread in Panel (b). An EDHEC-Risk Institute Publication 33

34 Appendix Figure 3: Simulated distribution of surplus attained with three LDI strategies that differ through the PSP. Each panel shows the distribution of the surplus after ten years with a LDI strategy invested either in (PSP 1, LHP 1), (PSP 2, LHP 1) or (PSP 3, LHP 1). The allocation to PSP 1 in the benchmark strategy is 50%, and the allocations to the other two PSPs are adjusted so as to keep the expected terminal wealth unchanged. These distributions have been obtained by generating 50,000 values for the surplus, assuming a log-normal distribution for the three asset classes (stocks, sovereign bonds, corporate bonds). The vertical red line represents the reference situation of a zero surplus. Also reported are the mean of the surplus (μ), its standard deviation (s), and the probability of a negative surplus ( Prob. ). Liabilities are discounted at the risk-free rate plus the spread, and the initial values are A 0 = L 0 = $1. 34 An EDHEC-Risk Institute Publication