An Empirical Analysis of the Benefits of Inflation-Linked Bonds, Real Estate and Commodities for Long-Term Investors with Inflation-Linked Liabilities

Transcription

1 An Empirical Analysis of the Benefits of Inflation-Linked Bonds, Real Estate and Commodities for Long-Term Investors with Inflation-Linked Liabilities Lionel Martellini and Vincent Milhau January 25, 2013 Abstract This paper proposes an empirical analysis of the welfare gains involved in introducing various assets with attractive inflation-hedging properties for long-term investors facing inflation-linked liabilities. Using formal intertemporal spanning tests, we find that interest rate risk dominates inflation risk so dramatically within instantaneous liability risk that introducing inflation-linked bonds, commodities or real estate within the liability-hedging portfolio has relatively little impact on investors welfare from a shortterm perspective. This holds true in spite of the attractive (in the case of real assets) and even perfect (in the case of inflation-linked bonds) inflation-hedging benefits of these asset classes. On the other hand, substantial welfare gains are obtained as the investment horizon converges towards the liability maturity date. Even more substantial utility gains are obtained if these asset classes are also used in the performance-seeking portfolio, where they provide diversification benefits with respect to bond and equity returns. Lionel Martellini is aprofessor of finance at EDHEC Business School and the Scientific Director of EDHEC- Risk Institute. Vincent Milhau is the Deputy Scientific Director of EDHEC-Risk Institute. The author for correspondence is Lionel Martellini. He can be reached at EDHEC-Risk Institute, 400 Promenade des Anglais, BP 3116, Nice Cedex 3 France. Ph: +33 (0) Fax: +33 (0) lionel.martellini@edhec.edu. This research has benefitted from the support of the Advanced Investment Solutions for Liability Hedging for Inflation Risk research chair, supported by Ontario Teachers Pension Plan. We are grateful to Bernd Scherer for useful comments, and to Andrea Tarelli for helpful research assistance. 1

2 1 Introduction A recent surge in inflation uncertainty has increased the need for investors to hedge against unexpected changes in price levels. Inflation hedging is a concern of particularly critical importance for pension funds, in situations when pension payments are indexed with respect to consumer price or wage level indexes. The implementation of inflation-hedging portfolios has become relatively straightforward in specific contexts where either cash instruments (inflation-linked bonds, such as Treasury inflation protected securities, or TIPS) or dedicated OTC derivatives (such as inflation swaps) can be used to achieve effective inflation hedging. More generally, however, the lack of capacity for inflation-linked cash instruments and the increasing concern over counterparty risk for derivatives-based solutions leave most investors with the presence of non-hedgeable inflation risk. In this context, a variety of financial asset classes (stocks and bonds) and real asset classes (commodities or real estate in particular) have been analyzed in terms of their ability to provide attractive inflation-hedging benefits. The results of such empirical investigations, however, have been mixed. On the one hand, starting with traditional asset classes, stock investments appear as relatively poor inflation hedging vehicles from a short-term perspective (see Fama and Schwert (1977), Fama (1981), Gultekin (1983) and Kaul (1987) among others). On the other hand, stocks appear to offer significant inflation protection over long horizons (Boudoukh and Richardson (1993) or Schotman and Schweitzer (2000)). This property is a priori particularly appealing for long-term investors such as pension funds, who need to match increases in price level at the horizon, but not necessarily on a monthly basis. A similar discrepancy between short-term and long-term inflation-hedging properties is expected for bond returns. Moving beyond stocks and bonds, recent academic research has also suggested that alternative forms of investments may offer attractive inflation-hedging benefits. Commodity prices, in particular, have been found to be leading indicators of inflation in that they are quick to respond to economy-wide shocks to demand (see e.g. Gorton and Rouwenhorst (2006)). In the same spirit, it has also been found that commercial and residential real estate provide at least a partial hedge against inflation, an effect that seems to be particularly significant over long-horizons (see Fama and Schwert (1977), Hartzell et al. (1987), Rubens et al. (1989) or Anari and Kolari (2002)). The implications of such findings for asset-liability management(alm) have been discussed by Hoevenaars et al. (2008), who construct optimal mean-variance portfolios with respect to inflation-driven liabilities. In this paper, we argue that what should matter to investors endowed with liability objectives is not the inflation-hedging property of an asset class, but rather its ability to hedge liability risk. The two concepts are in general not equivalent, and they actually coincide only at the date when liabilities are paid. Indeed, the present value of inflation-indexed liabilities is impacted by both inflation risk and interest rate risk, through the discount factor. It turns out 2

3 that for reasonable parameter values, short-term (instantaneous) liability risk consists mostly of interest rate risk, especially if liabilities have a long duration. As a consequence, for our calibrated parameter values, a liability-hedging portfolio solely invested in nominal bonds already spans close to 90% of the instantaneous variance of 5-year inflation-indexed liabilities, a percentage that grows to 95% or 97% when maturity is extended to 10 or 15 years. In view of the low residual variances, introducing assets such as inflation-linked bonds or real assets can merely increase by a few percents the proportion of the variance of the liabilities that can be hedged by the liability-hedging portfolio. It is only for short horizons (e.g., a year) that the relative importance of inflation risk versus interest rate risk becomes substantial within total liability risk. While inflation risk hedging is not a meaningful problem from the short-term perspective, one would indeed expect its importance to increase substantially while getting close to horizon. At maturity, of course, exposure to interest rate risk is zero, which leaves inflation risk remain as the sole source of uncertainty. 1 As a result, nominal bonds, while extremely efficient at hedging liability risk from the short-term perspective, turn out to be extremely ill-suited from the long-term perspective. Therefore, it remains to measure the utility gains achieved by introducing various asset classes with attractive (real estate and commodities) or even perfect (inflation-linked bonds) in the liability-hedging portfolio. Our paper provides a quantitative measure of these gains, in the context of a formal long-term investment model. Our paper is of course not the first to analyze the long-term investment problem in the presence of inflation risk. In a continuous-time model similar to ours, Brennan and Xia (2002) show by mathematical arguments that the introduction of inflation-indexed bonds increases investor s welfare. In a discrete-time VAR model, Campbell et al. (2003) show that investors with infinite risk aversion and recursive utility from consumption hold only perpetual real bonds. Campbell et al. (2009) argue that inflation-linked bonds allow investors endowed with real liabilities to reduce the long-term risk of their portfolio more substantially than other asset classes would do. On a different note, they also point out that inflation-indexed bond prices provide information that are useful to infer inflation expectations, which is of interest for monetary authorities. 2 Other papers have studied the benefits of adding real assets to a portfolio of stocks and bonds. Froot (1995) defines real assets as assets that tend to increase in nominal value in the face of inflation and notes that commodities are good at diversifying stocks and nominal bonds. Using a discrete-time VAR, Hoevenaars et al. (2008) examine the properties of alternative classes such as commodities and real estate. They report that commodities have 1 We assume away longevity risk in this discussion, as in the rest of the paper. 2 Althoughcluesaboutinflationexpectationsaboundinfinancialmarkets, inflation-indexedsecuritieswould appear to be the most direct source of information about inflation expectations and real interest rates (Governor Ben S. Bernanke before the Investment Analysts Society of Chicago, April 15, Also quoted in Grishchenko and Huang (2008)). 3

4 both attractive inflation-hedging properties and low correlation with stocks and bonds, which make them well-suited for diversifying a portfolio. We complement these papers in several directions. First, our analysis is done in a continuoustime model with stochastic investment opportunities which allows for an analytical derivation of the optimal strategy. This model and its solution are not new (see e.g. Munk et al. (2004)), but we use them to derive term structures of inflation and liability-hedging qualities. The advantage of a continuous-time model over a VAR model as in Campbell et al. (2003) is that it allows to incorporate no-arbitrage restrictions and involves only interpretable parameters, while coefficients of VAR models lack interpretability. Second, we provide for the first time, based on reasonable parameter values calibrated to US data, a quantitative estimate of the welfare loss induced by removing inflation-indexed bonds from the liability-hedging portfolio, as opposed to analyzing the question from an asset-only perspective. This allows us to perform an independent analysis of the benefits of inflation-linked bonds, commodities or real estate for liability hedging purposes versus diversification purposes. The computation of welfare gains is based on the strategy that is optimal in the presence of the stochastic opportunity set, so that we do not have to assume a fixed-mix. Third, we assess the benefits of introducing real assets such as real estate and commodities, which have attractive but imperfect, inflationhedging properties. We also provide a detailed analysis of the term structure of welfare gains, which sheds new light on the conflicting results obtained when addressing the question of liability-hedging from a short-term or a long-term perspective. The rest of the paper is organized as follows. In Section 2, we introduce the model and we calibrate it to market data. In Section 3 we analyze the welfare losses related to removing inflation-linked bonds from liability-hedging portfolios. In Section 4, we extend the analysis to real estate and commodities, which are found to have attractive, albeit imperfect, inflation hedging properties. Section 5 concludes and proposes a number of suggestions for further research. Technical details are relegated to a dedicated appendix. 2 The Model In this section, we present the model and the notation and we derive the optimal portfolio strategy. 2.1 Financial Variables Uncertainty in the economy is modeled by a standard probability space (Ω, A,P). The finite time span if [0,τ L ], where τ L is the maturity of liabilities. Innovations are modelled as the innovations to a 5-dimensional Brownian motion z. 4

5 The nominal short-term rate evolves as (Vasicek, 1977): dr t = a(b r t )dt+σ rdz t, (2.1) where z r is a Brownian motion. There exists a locally risk-free asset, the bank account, whose value is St 0 = e t 0 rsds. The zero-coupon yield of maturity τ is given by: y(t,τ) = D(τ) τ where b = b σrλr a, λ r is the price of interest rate risk, and: D(τ) = 1 e aτ a r t C(τ), (2.2) τ ], C(τ) = b[d(τ) τ]+ σ2 r [τ 2D(τ)+ 1 e 2aτ. (2.3) 2a 2a The investor has access to a constant-maturity bond index with maturity τ B = 10 years, whose price evolves as: db t B t = [r t D(τ B )σ r λ r ] dt D(τ B )σ rdz t. (2.4) In addition to the bond index, the investor can also trade in a stock index S with stochastic Sharpe ratio (Kim and Omberg, 1996): ds t = [ ] r t +σ S λ S t dt+σs dzt S, (2.5) S t dλ S t = κ ( ) λ λ S t dt+σ λ σ λdz t. (2.6) The real asset Y (real estate or commodity) evolves as: and the stochastic price index as: dy t Y t = [r t +σ Y λ Y ] dt+σ Y dz t, dφ t Φ t = πdt+σ Φdz t. (2.7) Liabilities are represented by an inflation-indexed zero-coupon bond(as in Hoevenaars et al. (2008) and Detemple and Rindisbacher (2008)), that is a zero-coupon that pays Φ τl at the maturity date τ L. From Martellini and Milhau (2012), the value of liabilities is: [ ] MτL L t = E t Φ τl = Φ t exp[ D(τ L t)r t +E(τ L t)], t τ L, (2.8) M t 5

6 with: E(u) = ( b σ ) rλ r [D(u) u]+(π σ Φ λ Φ )u σ rσ Φ ρ rφ a a + σ2 r 2a 2 [u 1 e au a ] [u 2 1 e au a ] + 1 e 2au. 2a More generally, we denote with I(t,τ) the price at date t of an indexed bond maturing at date τ. A straightforward application of Ito s lemma to (2.8) shows that the dynamics of L is: dl t L t = [r t D(τ L t)σ r λ r +σ Φ λ Φ ] dt D(τ L t)σ rdz t +σ Φdz t. (2.9) Hence the volatility vector of liabilities is σ L (τ L t) = D(τ L t)σ r +σ Φ. The volatility matrix σ collects the volatility vectors of all traded assets (except cash, which is locally risk-free). Thus, it depends on the investment universe. In this paper, we shall consider various investment universes, but all of them contain at least the nominal bond. The spanned price of risk vector λ also depends on the investment universe considered, but it can always be written as an affine function of the stochastic Sharpe ratio: λ t = Λ 1 +λ S t Λ 2, t 0. The market is in general incomplete, because of the presence of unspanned sources of risk (Sharpe ratio risk and/or inflation risk). Hence, there exist infinitely many price of risk vectors, which are those vectors of the form λ+ν, where ν satisfies σ ν t = 0 for all date t. Each of them gives rise to a pricing kernel: [ t M t = exp 0 (r s + λ ) s 2 + ν s 2 ds Optimal Portfolio Choice t 0 (λ s +ν s ) dz s ], t 0. (2.10) We now solve for the optimal portfolio policy for an investor facing the opportunity set described in section 2. The portfolio is self-financed, so the strategy is completely characterized by the vector w t that collects the weights allocated to the locally risky assets at time t. If the weights do not sum up to one, cash is used to make the balance. Thus, the value A of a portfolio evolves according to the following budget equation: da t A t = (r t +w tσ λ t ) dt+w tσ dz t. 6

7 The investor has horizon T, which we assume to be less than or equal to the maturity of liabilities (T τ L ). He is not explicitly concerned with asset value at date T, but instead with asset value relative to liability value L T. Different strategies are therefore compared on the basis of the expected utility from the terminal funding ratio A T /L T. In this paper, we will consider the Constant Relative Risk Aversion (CRRA) utility function: U(x) = x1 γ 1 γ, x > 0. With these specifications, the portfolio choice problem can be written as the problem of maximizing expected utility from terminal real wealth: [ ( )] max E AT U. (2.11) w L T To solve this problem, we follow the related literature that considers the problem of maximizing expected utility from real wealth (see e.g. Brennan and Xia (2002), Munk et al. (2004) and Sangvinatsos and Wachter(2005)). The solution technique is the martingale approach, which is based on the equivalence between the dynamic problem (2.11) and a static problem where the control variable is the terminal wealth. As shown by He and Pearson (1991), in an incomplete market, the solution involves the minimax pricing kernel, which is the pricing kernel obtained with a special ν denoted ν in (2.10). The following proposition gives a description of the solution. The derivation of a similar result can be found in Martellini and Milhau (2010). Proposition 1 wt = λpsp t where: γσt PSP The optimal portfolio rule is given by: ( wt PSP ) ( βt L wt LHP 1 1 γ γ ) [C2 ] (T t)+c 3 (T t)λ S t β λ w λ, wt PSP 1 = (σ σ) 1 σ λ 1 (σ σ) 1 σ t, λ t wt LHP 1 = 1 (σ σ) 1 σ σ L (τ L t) (σ σ) 1 σ σ L (τ L t), w λ 1 = σ) 1 σ σ λ, 1 (σ σ) 1 σ σ λ(σ The functions C 1, C 2 and C 3 are solutions to ordinary differential equations: Ċ 3 = 1 γ γ (2.12) [ [ ] σ 2 λ (1 γ)σ λnσ λ C κ 1 γ γ σ λλ 2 ]C 3 + Λ 2 2, (2.13) γ 7

8 Ċ 2 = Λ 2σ L (τ L T + )+ Λ 2[Λ 1 σ L (τ L T + )] γ + 1 γ [ ] σ 2 γ λ (1 γ)σ λnσ λ C2 C 3 + [ κλ+ 1 γ γ σ λ[λ 1 σ L (τ L T + )]+ (1 γ)2 γ [ κ 1 γ ] γ σ λλ 2 C 2 ] σ λnσ L (τ L T + ) C 3, (2.14) Ċ 1 = D(τ L T + )σ r λ r σ Φ λ Φ +Λ 1σ L (τ L T + )+ Λ 1 σ L (τ L T + ) 2 + [ κλ+ 1 γ (1 γ)2 σ L (τ L T + ) Nσ L (τ L T + )+ 1 2γ 2 σ2 λc 3 [ ] σ 2 λ (1 γ)σ λnσ λ C γ 2γ γ σ λ[λ 1 σ L (τ L T + )]+ (1 γ)2 γ 2γ ] σ λnσ L (τ L T + ) C 2, (2.15) with N being the matrix of the projection onto the orthogonal space of the columns of σ: N = I σ(σ σ) 1 σ. The decomposition (2.12) is standard, in view of the results of Detemple and Rindisbacher (2010), who establish a general separation formula for optimal portfolios, and of those of Munk et al. (2004), who study a model close to ours. The first building block is the performanceseeking portfolio (PSP). The allocation to this block is increasing in its Sharpe ratio, and decreasing in its volatility and in the risk aversion. The second building block is the liabilityhedging portfolio (LHP), which, by construction, has the highest squared correlation with innovations to the liability value. This maximum correlation is equal to one only if liability risk is spanned, which is equivalent here to saying that inflation risk is spanned (interest rate risk is already spanned by the nominal bond). The allocation to the LHP is increasing in the risk aversion, which makes sense since risk-averse investors seek to reduce the volatility of their funding ratio. It is also increasing in the beta of liabilities with respect to the LHP, a parameter that measures how well the LHP replicates liabilities. The third building block is the Sharpe ratio-hedging portfolio, that is the portfolio that best hedges unexpected changes in the Sharpe ratio. The allocation to this block is decreasing, as opposed to increasing, in the beta, since the investor prefers to hold assets that are negatively correlated with the Sharpe ratio. In practice, this portfolio is mostly invested in stocks, because of their strongly negative correlation with Sharpe ratio (see Barberis (2000), Xia (2001) and our own calibration in Subsection 2.4 below). 8

9 2.3 Expected Utility for the Class of Affine Strategies In the empirical sections (3 and 4) of this paper, we will compute the expected utility of various portfolio strategies based on different investment universes. For computational efficiency, it is of high interest to have closed-form expressions for these expected utilities. 3 As shown in the next proposition, quasi-analytical expressions can be derived for strategies such that the volatility vector of wealth is affine in the Sharpe ratio λ S : σw t = H 1 (T t)+λ S t H 2 (T t), (2.16) where H 1 and H 2 are functions of the time-to-horizon T t. Note that the optimal strategy (2.12) fits into this category. The next proposition gives a quasi-analytical expression for the expected utility. It extends a similar result established by Sangvinatsos and Wachter (2005) to the case where not all state variables are spanned by traded assets. Proposition 2 Assume that the vector of weights is of the form (2.16). Then the expected utility from terminal funding ratio is given by: E t [U ( AT L T )] = U ( At L t ) h(t,λ S t ) γ, where: [ [ 1 γ h(t,λ S t ) = exp E 1 (T t)+e 2 (T t)λ S t + 1 γ 2 E 3(T t) ( ]] ) λ S 2 t, and the functions E 1, E 2 and E 3 are solutions to the following ODEs: E 3 = (1 γ)σ 2 λe [(1 γ)σ λh 2 κ]e 3 +2H 2Λ 2 γ H 2 2, E 2 = H 1Λ 2 +H 2[Λ 1 +(1 γ)σ L (τ L T + )] γh 1H 2 +(1 γ)σ 2 λe 2 E 3 + [ κλ+(1 γ)[h 1 σ L (τ L T + )] σ λ ] E3 +[ κ+(1 γ)h 2σ λ ]E 2, 3 In the absence of such expressions, expected utility can be computed with Monte-Carlo simulations, which increases computation time. 9

10 E 1 = D(τ L T + )σ r λ r σ Φ λ Φ +H 1Λ 1 (1 γ)h 1σ L (τ L T + ) γ 2 H γ 2 σ L (τ L T + ) γ σ 2 2 λe σ2 λe 3 + [ κλ+(1 γ)σ λ[h 1 σ L (τ L T + )] ] E 2. This result will allow us to compute the Monetary Utility Loss of using a strategy (a) as opposed to a strategy (b): by definition, the MUL is the capital that must be invested in (a) in addition to A 0 in order to achieve the same expected utility as by investing A 0 in (b). It follows from the previous proposition that the MUL can be computed in quasi-closed form, and is given by: MUL A 0 = [ ] h (b) (0,λ S 0) γ 1 γ h (a) (0,λ S 0) In particular, the ratio MUL A 0 is independent from A Model Calibration 1. (2.17) We calibrate the model to quarterly US data. For the nominal term structure, we use nominal yields on 3-month T-bills and on Government Bonds of maturities 1 year, 3 years, 5 years and 10 years, which are available from Bloomberg over the period Q to Q The US Consumer Price Index for All Urban Consumers is obtained from the Federal Reserve Economic Database, for the period from Q to Q The stock index is represented by the S&P500, which is available from CRSP over the period Q to Q Since the Sharpe ratio is not directly observable, we relate it to an observable predictor of stock returns, as is usual in the literature on stock return predictability. Following Campbell and Viceira (1999), we assume that the expected excess return on the index is an affine function of the dividend yield, computed as the sum of dividends paid over the last four quarters, divided by the current index value: [ E t log S ] t+ t ( t)y(t, t) = mdy t +p, (2.18) S t It can be shown that this equality implies an affine relationship between the Sharpe ratio and the dividend yield, λ S t = mdy t + p. Finally we consider real estate and commodities in the real asset class. The real estate index used is the National Council of Real Estate Investment Fiduciaries Property Index (NCREIF), available from Bloomberg from Q to Q1.2011: a non-reits index has been chosen so as to avoid artificial exposure to the stock market. Commodities are represented by the S&P Goldman Sachs Commodity Index (GSCI), available from Q to Q We estimate model parameters by maximizing the likelihood of observations, except for 10

11 the price of inflation risk, λ Φ, which we estimate from breakeven inflation rates: BEI(t,τ) = y(t,τ) y I (t,τ), where y I (t,τ) denotes the real yield of maturity τ. Real yields with maturities τ = 10, 20 and 30 years are available from Bloomberg over Q to Q It follows from the expression of an indexed zero-coupon price (see e.g. Martellini and Milhau (2012)) that: BEI(t,τ) = π σ Φ λ Φ σ rσ Φ ρ rφ a The price of inflation risk can then be approximated as: λ Φ 1 σ Φ 1 n [1 1 e aτ aτ n [BEI(t i,τ) π ti ], i=1 and we take the University of Michigan Inflation Expectation, available in the Federal Reserve Economic Database, as a proxy for the expected inflation rate π ti. This implies λ Φ =65.25% for τ = 5 years, 61.10% for τ = 10 years and 53.33% for τ = 30 years. We retain a value of 60%. Results are shown in Table 1, together with the standard errors of estimators. Unsurprisingly, parameters that relate to risk premia (λ r, λ Y, m and p) are estimated with relatively low precisions, with standard deviations representing approximately 30% to 100% of the point estimate. Figure 1 shows the Sharpe ratio implied by the postulated relation between expected returns and the dividend yield. ]. 3 Is Inflation Risk Prominent in Liability Risk? The volatility vector of liabilities can be broken down into a contribution of interest rate and a contribution of inflation: σ L (τ L t) = D(τ L t)σ r +σ Φ. In this section we give a quantitative measure of the relative importance of interest rate risk and inflation risk. 3.1 Short-Term Liability Risk To illustrate that interest rate risk is dominant from a short-term perspective when the maturity of liabilities is sufficiently large, we compute absolute value of the instantaneous correlation 11

12 between LHP and liabilities, which is the ratio: σ(σ σ) 1 σ σ L (τ L t). (3.1) σ L (τ L t) The square of the ratio(3.1) can be interpreted as a percentage of total variance of liabilities that is explained by the traded assets. In Table 2 we consider various investment universes, which lead to different LHPs. In this table and in the following figures, I denotes an inflationlinked of constant maturity equal to ten years. As this bond has a constant duration, it does not perfectly replicate the value of liabilities, which have a decreasing duration. We also consider different values for the time to maturity τ L t, namely 1, 5, 10 and 15 years. It turns out that the correlation is already greater than 98% for a LHP of nominal bonds only and a horizon of five years or more. In these conditions, the marginal benefits of incorporating stocks and/or real assets into the LHP can only be marginal: at most 10 bp of correlation are gained. Another salient feature is that asset classes such as equity, real estate and commodities have low instantaneous correlations with liabilities: they are lower than 10% for stocks for all maturities, and they slightly exceed 50% only for real estate. It is clear from the numbers reported that nominal bonds are needed in order to attain a high correlation with liabilities. A perfect instantaneous correlation can be achieved in only two situations: either the LHP contains only an inflation-linked bond with the same maturity as the liability, or the LHP contains both an indexed bond of any maturity, and a nominal bond. Indeed, if the maturity of liability differs from that of the indexed bond, a duration adjustment is needed so as to match the respective exposures to interest rate risk of the LHP and the liabilities. 3.2 Long-Term Analysis As long-term correlations between LHPs and liabilities can differ substantially from short-term ones, we now turn to a long-term analysis. We consider the following portfolio strategy, that is obtained by deleting the demands for the PSP and for the Sharpe ratio-hedging portfolio in (2.12): w t = β L t w LHP t. The correlation between the wealth generated by this strategy and liabilities, to which we shall simply refer as the correlation between the LHP and liabilities, can be derived in closed form for various investment horizons. Figure 2 displays the term structure of correlations for four different LHPs (nominal bonds only, and nominal bonds with either real estate, commodities or inflation-indexed bonds), and the two extreme maturities for liabilities (1 year and 15 years). Unsurprisingly, the correlation increases in the number of assets, because the relative importance of the unhedgeable part in liability risk decreases. When inflation-linked bonds 12

13 are included together with nominal bonds, a perfect correlation is achieved at all horizons. For imperfect LHPs, the correlation is always much larger at short horizons than at maturity. It is also interesting to note that for liability maturities of five to fifteen years, the correlation stays between 90% and 100% for horizons comprised between one quarter and about two thirds of the maturity, and then rapidly decreases. Then it reaches a minimum, and then starts to grow just before maturity, but the increase does not make up for the previous decrease and the correlation levels observed at short horizons are never recovered at maturity. Overall, it turns out that a LHP that does not contain inflation-indexed bonds has a rather low correlation with liabilities, always less than 50%, over investment periods that have the same length as the maturity of liabilities. 3.3 Utility Cost of Imperfect Inflation Hedging The correlation between wealth and liabilities provides a measure of the fraction of longterm liability risk that is hedged by traded assets, but it does not fully represent investor s preferences. Instead, the utility of the terminal funding ratio A T /L T measures the welfare generated by a given allocation strategy. To assess the welfare loss induced by the impossibility to hedge completely liability risk, we compute the Monetary Utility Loss (MUL) of a strategy that employs only nominal bonds or only nominal bonds and stocks in the LHP, with respect to a strategy that also uses inflation-indexed bonds. In order to isolate the benefits of inflation-indexed bonds from a liability-hedging perspective, we keep the composition of the PSP and of the Sharpe ratio-hedging portfolio constant. Formally, we denote with Wt PSP and W λ these portfolios when they are computed with nominal bonds and stocks only, and we consider two LHPs: W LHP,(a1) t is invested in nominal bonds and stocks only, while W LHP,(b1) t also contains indexed bonds. The compositions of these portfolios are obtained by using the expressions of Proposition 1, where the volatility matrix σ and the price of risk vector λ are adapted to the investment universe. We then consider strategies (a1) and (b1): wt i = λpsp t γσt PSP ( Wt PSP ) β L,i γ t W LHP,i t ( 1 1 γ ) [C2 ] (T t)+c 3 (T t)λ S t β λ W λ, (3.2) where i = (a1) or (b1). Note that the beta of liabilities with respect to the LHP depends on the LHP considered. 4 We compute the MUL of strategy (a) with respect to (b), using (2.17). Since the utility cost depends on the risk aversion γ, which is not observable, we set 4 Functions C 2 and C 3 are obtained by numerically solving the ODEs of Proposition 1. In order to have the same values of C 2 (T t) and C 3 (T t) for both strategies (a) and (b), we solve the ODEs under the assumption that the investment universe contains the three asset classes (recall that the investment universe has an impact on C 2 and C 3 through the matrices Λ 1, Λ 2 and N). 13

14 this parameter so as to achieve target allocations to the PSP at date 0 of 75%, 30% and 15% respectively: the implied (rounded) values of γ are 6, 15 and 30. The corresponding allocations to stocks at date 0, which can also be computed from (2.12), are %, 68.51% and 36.96%. Utility losses of removing indexed bonds from the LHP are reported in Figure 3. The MUL appears to be increasing in the risk aversion parameter, which happens for two reasons. First, with a higher γ, the investor allocates more to the LHP. Any change in the composition of the LHP will thus have a greater impact. Second, a higher γ means that the investor penalizes more the uncertainty in the final funding ratio. The MUL also appears to be increasing in the investment horizon, which is a straightforward effect since a longer horizon magnifies the effects of imperfect liability hedging. Overall, the MULs reported in Figure 3 do not seem prohibitively high over short-horizons, but more substantial welfare gains/losses are obtained as the time-horizon converges towards the liability maturity date. For example, if γ = 30 (which corresponds to a conservative initial allocation of about 15% in the PSP), the MULs are less than 2% of initial wealth for a liability maturity of one year, but they reach 24% at horizon for a maturity of 15 years. In an attempt to assess the economic significance of these utility costs, we have computed, for comparison purposes, the cost of ignoring predictability in stock returns. In this exercise, we have assumed that the investor has access only to nominal bonds, stocks and cash, and has the choice between the following two options: (a2) ignore the predictability in excess stock returns and treat λ S as if it was a constant parameter equal to λ; (b2) follow the strategy that is actually optimal in the presence of stochastic interest rate and stochastic equity risk premium, as described in Proposition 1. With the former strategy, the hedging demand against equity risk premium risk becomes zero, and the PSP is a fixed-mix portfolio. Figure 4 shows the MULs for different investment horizons and different levels of risk aversion. Of course, the utility cost of treating the Sharpe ratio as a constant is virtually zero for a horizon as short as one year, because mean reversion is a long-term phenomenon. Moreover, this cost is decreasing in the risk aversion because ignoring predictability hurts less investors who have only a small fraction of their holdings invested in stocks. Comparing these utility losses with those induced by imperfect liability hedging, it can be seen that the utility cost of ruling out inflation-indexed bonds from the LHP is larger than that of ignoring predictability for highly risk-averse investors (γ = 30) and a liability maturity of 1 year. For the 15-year liabilities, the utility cost of imperfect hedging is larger, except when the investment horizon approaches 15 years, but the two costs at the horizon are similar to each other (24.03% for imperfect hedging vs 26.16% for predictability). On the other hand, for little risk averse investors, who initially invest 50% or 75% of their wealth in the PSP, the utility cost of ignoring predictability is larger. These results show that perfectly hedging liability risk is more important than taking into account predictability for highly risk-averse investors, but that it is less important for 14

15 investors with low or medium risk aversion. 4 Benefits of Real Assets for Liability Hedging and Diversification Purposes In this section, we measure the benefits of real estate and commodities both from a liabilityhedging perspective and from a global perspective, that encompasses both their diversification properties and their ability to hedge inflation risk and equity premium risk. 4.1 Utility Cost (Gain) of Excluding (Including) Real Assets from (in) the LHP We first measure the utility cost of excluding real assets from the LHP, in the same way as we computed the utility cost of excluding indexed bonds in Subsection 3.3. We still assume that the PSP and the Sharpe ratio-hedging portfolio are invested in nominal bonds and stocks only (portfolios denoted with Wt PSP and W λ in Subsection 3.3). But there are two versions of the LHP: a LHP of nominal bonds and stocks only, denoted with W LHP,(a3) t, and a LHP invested in nominal bonds, stocks and one real asset class (commodities or real estate), denoted with W LHP,(b3) t. The three building blocks are combined as in (3.2). Figure 5 shows the MUL of strategy (a3) with respect to (b3), that is the utility cost of removing commodities from the LHP. Figure 6 reports similar results for the real estate class. These utility losses never exceed 3% of initial wealth, and are in all cases significantly smaller than those reported in Figure 3. Hence the marginal gains of adding real assets to a LHP of nominal bonds and stocks are far smaller than those of including inflation-linked bonds in the LHP. As a result, a LHP composed of nominal bonds, stocks and commodities or real estate does not represent a significant improvement over a LHP made of nominal bonds and stocks only. 4.2 Utility Cost (Gain) of Excluding (Including) Real Assets from (in) All Building Blocks Beyond their usefulness, or lack thereof, for liability-hedging purposes, real assets may also be useful in the performance-seeking component of investors portfolios due to their positive risk premia and their relatively low correlations with stocks and bonds, which makes them attractive for diversification purposes. To investigate more formally the benefits of including real assets in the PSP, we compute the utility loss of the optimal strategy that uses nominal bonds and stocks with respect to the optimal strategy invested in these two classes plus one 15

16 real asset class. This MUL aggregates the benefits of real asset in terms of liability hedging, diversification (within the PSP), as well as possibly Sharpe-ratio risk hedging within the dedicated intertemporal hedging demand. Results are shown in Figures 7 and 8. A first observation is that the MUL is now higher for more aggressive investors (γ = 6 or 15) than for conservative ones (γ = 30), which contrasts with Figure 3, where the MUL was increasing in the risk aversion. In other words, the MUL of excluding commodities or real estate from the investment universe is higher for investors who allocate a significant fraction of wealth (here 75% or 50%) to the PSP than for those who invest mainly in the LHP (15% in the PSP). This result suggests that the benefits of real assets are in fact to be found in the PSP more than in the LHP. For liabilities of maturity 15 years, the least risk-averse investor must invest 13% more in stocks and bonds to achieve the same expected utility as with these two classes plus commodities after 15 years, and 23% more to achieve the same expected utility as with the two traditional classes and real estate. These utility costs are substantial, although they are still much lower than the utility cost of ignoring predictability, which reaches 91.6% for the same investor. Nevertheless, they are much higher than the utility costs of excluding real assets from the LHP only, which hardly reach 3%. These results therefore suggest that real assets provide benefits to long-term investors, even in situations where their usefulness is limited in terms of inflation and liability hedging purposes. 5 Conclusion This paper proposes an empirical analysis of the opportunity costs involved in removing various financial and real assets with attractive inflation-hedging properties for long-term investors facing inflation-linked liabilities. We find that instantaneous liability risk is srongly dominated by interest rate risk, due to the discount factor, so that a liability-hedging portfolio of nominal bonds only provides already a very high correlation with liabilities. In this context, the gain in correlation of introducing inflation-indexed bonds or real assets is only marginal. On the long run, the perspective is different, since inflation risk becomes relatively more important, and in fact remains the only source of uncertainty at the date liabilities are paid off. Nevertheless, we find that for reasonable parameter values, the utility cost of excluding inflation-indexed bonds from the liability-hedging portfolio is lower than that of ignoring predictability in stock returns, except for very conservative investors. Finally, we analyze the utility cost of excluding real estate and commodities either from the liability-hedging portfolio only, or from all building blocks. The cost of giving up real assets in the liability-hedging portfolio only is found to be very low, but the cost of excluding them from all building blocks appears to be much more substantial. The added value of these asset classes thus appears to come mainly from their diversification benefits with respect to equities and bonds. 16

17 Our work could be extended in several dimensions. One important extension would be to analyze how our results might be affected by the introduction of a stochastic expected inflation component. One key concern relates in particular to a possible increase in expected inflation, which would lead to a drop in nominal bond prices without a corresponding impact on liability values, thus weakaning the performance of nominal bonds in hedging short-term rrisk for longterm inflation-linked liabilities. 5 Finally, the analysis in this paper does not tell us what would be the marginal welfare impact of improving inflation/liability hedging properties of asset classes that are substantial ingredients in the performance-seeking portfolio. In particular, we leave for further research an analysis of the welfare benefits of equity portfolios constructed to have improved long-term liability hedging properties (see for example Ang et al. (2012) for an analysis of the crosss-sectional dispersion in inflation betas within the equity universe). 5 A long-only position in nominal bonds will always have a negative exposure to changes in unexpected inflation. On the other hand, it is possible in principle to design a long-short nominal bonds portfolio strategy that achieves zero exposure to changes in unexpected inflation, while having an exposure to changes in real rate equal to that of the liabilities. Implementing such portfolio strategies, however, may not be feasible for investors subject to no shortsales constraints. 17

18 Table 1: Maximum Likelihood estimates. Nominal short-term rate: dr t = a(b r t )dt+σ rdz t. Estimate. Standard deviation. a b σ r λ r Stock price process: ds t /S t = [ ] r t +σ S λ S t dt+σ S dz t. Estimate. Standard deviation. σ S Sharpe ratio: dλ S t = κ ( ) [ λ λ S t dt+σ λ dz t, E t log St+ t S t ] ( t)y(t, t) = mdy t +p. Estimate. Standard deviation. κ λ σ λ m p Price index process: dφ t /Φ t = πdt+σ Φ dzφ t. Estimate. Standard deviation. π σ Φ λ Φ Real asset process: dy t /Y t = ( r t +σ Y λ Y) dt+σ Y dz t. Real estate. Commodities. Estimate. Standard deviation. Estimate. Standard deviation. σ Y λ Y Correlations. Estimate. Standard deviation. ρ rs ρ rλ ρ rφ ρ Sλ ρ SΦ ρ λφ Real estate. Commodities. Estimate. Standard deviation. Estimate. Standard deviation. ρ SY ρ ry ρ λy ρ ΦY Volatilities of residuals in bond yields. Estimate. Standard deviation. s 1Y < s 3Y < s 5Y < s 10Y < Estimates for all parameter values except λ Φ have been obtained by maximizing the log-likelihood of quarterly observations on US market. Standard deviations have been computed by taking the square roots of diagonal elements of the inverse of Fisher s information matrix.

19 Table 2: Decomposition of liability risk into spanned risk and unspanned risk. Time-to-maturity (years) LHP Corr. % Var. Corr. % Var. Corr. % Var. Corr. % Var. B S 0.43 < < < Com RE I B, S B, Com B, RE B, S, Com B, S, RE B, I This table shows the absolute value of the correlation between LHP and liability (column Corr.), as well as the percentage of the total variance of liabilities that is explained by the base assets (column % Var.). Eleven LHPs are considered, based on the following assets: nominal bonds of constant maturity 10 years (B), stocks (S), commodities (Com), real estate (RE) and indexed bonds of constant maturity 10 years (I). Parameter values are taken from Table 1. 19

20 Figure 1: Implied Sharpe ratio and total return index. 200 Sharpe ratio (%) Sharpe ratio Index Total return index ($) 80 Q Q Q The instantaneous Sharpe ratio of the S&P 500 index is estimated from the dividend yield, with the relationship λ S t = mdy t + p, where m and p are estimated by analyzing quarterly data over the period Q to Q The total return index series of the of S&P 500 is also obtained from CRSP. Its value is taken equal to $ 1 on January 1st, Figure 2: Term structures of correlation of liability-hedging portfolios with liabilities. (a) Liability maturity 1 year. (b) Liability maturity 15 years. Correlation B B, RE B, Com B, I 1/4 1/2 3/4 1 Horizon (years) Correlation B B, RE B, Com B, I Horizon (years) This figure shows the model-implied correlations between the wealth generated by a strategy that fully invests in the LHP and the cash, and the liabilities. Liabilities are represented by an inflation-indexed bond with initial maturity τ L equal to 1 or 15 years. The LHP is computed over one of the following four investment universes: nominal bonds and stocks (B, S); nominal bonds and real estate (B, RE); nominal bonds and commodities (B, Com); nominal bonds and inflation-indexed bonds (B, I). Parameters are set to the calibrated values (see Table 1). 20

21 Figure 3: Utility cost of giving up indexed bonds in the liability-hedging portfolio. (a) Liability maturity 1 year. (b) Liability maturity 15 years MUL (%) γ = 6 γ = 15 γ = 30 1/4 1/2 3/4 1 Investment horizon (years) MUL (%) γ = 6 γ = 15 γ = Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial capital must be increased for the investor to be willing to give up inflation-indexed bonds in the LHP, which leaves only stocks and nominal bonds in the LHP. Liabilities are represented by an inflation-indexed bond with initial maturity τ L equal to 1 or 15 year(s), and the investment horizon varies from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial values r 0 and λ 0 are taken equal to the long-term means b and λ. Figure 4: Utility cost of ignoring predictability in stock returns. (a) Liability maturity 1 year. (b) Liability maturity 15 years γ = 6 γ = 15 γ = γ = 6 γ = 15 γ = 30 MUL (%) MUL (%) /4 1/2 3/4 1 Investment horizon (years) Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial capital must be increased for the investor to be willing to ignore time variation in the Sharpe ratio of the stock index. The suboptimal strategy assumes a constant Sharpe ratio and ignores the hedging demand against changes in this parameter. Liabilities are represented by an inflation-indexed bond with initial maturity τ L equal to 1 or 15 year(s), and the investment horizon varies from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial values r 0 and λ 0 are taken equal to the long-term means b and λ. 21

22 Figure 5: Utility cost of excluding commodities from the liability-hedging portfolio. (a) Liability maturity 1 year. (b) Liability maturity 15 years. MUL (%) γ = 6 γ = 15 γ = 30 1/4 1/2 3/4 1 Investment horizon (years) MUL (%) γ = γ = 15 γ = Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial capital must be increased for the investor to be willing to replace a LHP of nominal bonds, stocks and commodities by a LHP of nominal bonds and stocks only. Liabilities are represented by an inflation-indexed bond with initial maturity τ L equal to 1 or 15 year(s), and the investment horizon varies from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial value λ S 0 is taken equal to the long-term mean λ. Figure 6: Utility cost of excluding real estate from the liability-hedging portfolio. (a) Liability maturity 1 year. (b) Liability maturity 15 years MUL (%) MUL (%) γ = 6 γ = 15 γ = 30 1/4 1/2 3/4 1 Investment horizon (years) 0.5 γ = 6 γ = 15 γ = Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial capital must be increased for the investor to be willing to replace a LHP of nominal bonds, stocks and real estate by a LHP of nominal bonds and stocks only. Liabilities are represented by an inflation-indexed bond with initial maturity τ L equal to 1 or 15 year(s), and the investment horizon varies from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial value λ S 0 is taken equal to the long-term mean λ. 22

23 Figure 7: Utility cost of excluding commodities from all building blocks. (a) Liability maturity 1 year. (b) Liability maturity 15 years γ = 6 γ = 15 γ = γ = 6 γ = 15 γ = /4 1/2 3/4 1 Investment horizon (years) Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial wealth must be increased for the investor to be willing to exclude commodities from the investment universe when he has the opportunity to invest in nominal bonds, stocks, commodities and cash. We consider three levels of risk aversion γ. The liability portfolio is a fixed-maturity indexed bond of initial maturity τ L equal to 1 or 15 year(s), and we let the investment horizon vary from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial value λ S 0 is taken equal to the long-term mean λ. Figure 8: Utility cost of excluding real estate from all building blocks. (a) Liability maturity 1 year. (b) Liability maturity 15 years γ = 6 γ = 15 γ = γ = 6 γ = 15 γ = /4 1/2 3/4 1 Investment horizon (years) Investment horizon (years) Monetary utility loss (MUL) is expressed as the percentage by which initial wealth must be increased for the investor to be willing to exclude real estate from the investment universe when he has the opportunity to invest in nominal bonds, stocks, commodities and cash. We consider three levels of risk aversion γ. The liability portfolio is a fixed-maturity indexed bond of initial maturity τ L equal to 1 or 15 year(s), and we let the investment horizon vary from one quarter to τ L. We consider three values for the risk aversion γ. Other parameters are set to the calibrated values (see Table 1), and the initial value λ S 0 is taken equal to the long-term mean λ. 23