*The following is extracted from “Alternative Investments for Institutional Investors: Risk Budgeting Techniques in Asset Management and Asset-Liability Management.” For further details on how to obtain a full copy, please see the end of this article.*

Allocation decisions dependent on the investment zone have been widely studied in the literature over the last decade. Brandt (2005) and Campbell and Viceira (2005) discuss the differences between short-term or myopic and intertemporal asset allocation decisions. The term structure of risk, driven solely by the presence of mean-reversion effects, with different speeds of mean reversion (Lettau and Wachter 2007), also plays a central role in asset allocation decisions in the presence of liabilities (see Campbell and Viceira 2005 for the notion of term of structure of risk). This article uses VECM model-implied dynamics to assess inflation hedging potential across different investment horizons.

Consistent with the portfolio separation theorem, we will study the liability-hedging portfolio (LHP) separately from the performance-seeking portfolio (PSP). In a framework where liabilities are indexed with respect to inflation, and when short-term liability risk hedging is the sole focus, the optimal LHP allocation consists of investing 100% in the inflation-indexed bond portfolio (TIPS portfolio), which unfortunately leads to very limited upside potential.

Consequently, the investor needs a relatively sizeable allocation to the PSP to meet the return requirements, which in turn generates a relatively high funding risk. Intuitively, one would expect that relaxing the constraint of a perfect liability fit for the LHP at the short-term horizon would allow one to include alternative asset classes in the LHP, which in turn leads to increased upside potential. Overall, this would allow an investor to reduce his/her allocation to the PSP, which leads to a reduced surplus risk.

To formalise this intuition, we perform a scenario-based analysis to derive the funding ratio distribution at various investment horizons. The data generating process is described by the vector error correction model (VECM) introduced in our full report (see weblink at the end of this article). We further use the structural model so as to disentangle the correlated innovation process and transform it into i.i.d. innovations. We draw i.i.d. random variables from the multivariate standard normal distribution for the structural innovations s {s = 1…S} and obtain the modelled returns by:

for a total of S = 5000 simulated paths. The first variable in Yst represents the liability return. We evaluate the different portfolios in terms of the funding ratio {FR} distribution. The funding ratio at t in scenario s is accordingly given by:

where denotes the n x 1 vector containing a 1 in the first position and zeros elsewhere, and is the portfolio vector. We first analyse the potential of stand-alone inflation-hedging portfolios before constructing optimal portfolios.

**Stand-alone hedging potential**

This article assess the inflation hedging potential of the various asset classes on a stand-alone basis. The analysis follows the methodology previously described, that is, all conclusions drawn from the funding ratio distributions over the 5000 simulated scenarios based on the VECM dynamics.

We compare investments in the perfect liability-hedging portfolio (TIPS) to investments in traditional assets (bonds and stocks) and to investments in alternative investments (commodities and real estate). Various investment horizons from three through 30 years are considered.

Table 1 presents the relevant indicators for the funding ratios assuming a 100% investment in the corresponding traditional or alternative asset class. The first column of the table refers to the liability-hedging portfolio that is obtained by 100% investment in an inflation-indexed security that has the same maturity as the liabilities.

Note that in reality such portfolios may be unavailable as TIPS are issued for a small number of maturities. For instance, in the US, only TIPS with maturity up to 10 years are available. Note further that we omitted statistics for T-bill portfolios from these tables. In fact, even though T-bills are well-correlated with the liability returns, they exhibit a significant lack of relative performance due to the term spread risk premia contained in the liability return. As a result, T-bills underperform liabilities significantly and are not a natural candidate for the liability-hedging portfolio. In this simplified setting, this 100% TIPS liability hedging portfolio in fact proves to be a perfect match for the liability, and therefore mean funding ratios are equal to 1 with a 100% probability for all time horizons.

In practice, the presence of non-financial sources of risk – e.g. actuarial risk – implies that there is some remaining funding risk even with a solution invested 100% in inflation-hedging instruments with maturities matching the maturity dates of the pension payments.

The mean funding ratios for other classes in Table 1 are higher than 1, which indicates that all assets have, on average, higher returns than the liability stream. This is consistent with the observed historical values that have been used to calibrate the data generating process described in the previous section. On the other hand, these asset classes involve the introduction of some liability risk, as evidenced by the number in panel B and panel C. The shortfall probabilities in panel B illustrate the time-horizon characteristics of the assets with respect to the liabilities. Indeed, shortfall probabilities systematically decrease as the investment horizon increases. This is consistent with results obtained for the model-implied volatilities and correlations with the liabilities (see Figures 1 & 2).

The correlation between bonds, stocks and real estate with the liabilities increases with the time horizon. Additionally, volatilities decrease with the investment horizon for bonds, stocks, and commodities, and slightly increase, in relative terms, for real estate, while the model-implied volatilities of liability returns sharply rise as the investment horizon increases.

Furthermore, the superior returns of the assets explain a part of the observed downwards sloping shortfall probabilities since they translate into a steeper positive trend in the numerator than in the denominator of the funding ratio. The strong decrease in shortfall probability for stocks is also explained by the structural relationship between liability shocks and aggregated responses to stock returns (see Figure 3).

Indeed, the persistence of liability shocks is much more pronounced for stocks than for other assets. As far as commodities are concerned, the response to liability shocks is immediate but not persistent, which explains why shortfall probabilities decrease less sharply than in the case of stocks. Real estate, on the other hand, reacts negatively in the short run, but ‘recovers’ as the liability shocks lead to persistently positive shocks to future real estate returns.

Overall, these shortfall probabilities are quite high in the short run, with numbers ranging from 37% (stocks and real estate) to 47% (long bond) and fall, at least for stocks and real estate, which suggests that moving away from TIPS, if it allows for better performance (mean funding ratios greater than 1) involves significant short-term liability risk. On the other hand, these values eventually decrease in the long run for stocks and real estate (with shortfall probabilities equal to 10% and 15% respectively), while they remain high for commodities (25%) and the long bond (38%).

In panel C (Table 1) we present the probabilities that the asset portfolio value falls “severely” short of the liability portfolio value. For short investment horizons, these severe or extreme shortfall probabilities are alarmingly high with numbers greater than 20%. Additionally, for the long bond, extreme shortfall probabilities do not seem to decrease in the long run (26% for three and for 30 years with even higher numbers in the medium term).

As far as commodities are concerned, one obtains a modest decrease from 26% (three to seven years) to 20% (30 years). On the other hand, stocks and real estate exhibit similar significantly downwards sloping patterns as in the case of shortfall probabilities (panel B). Across all four assets, we observe that the level of standard shortfall probabilities (panel B) which would obviously be of great concern to investors such as pension funds.

As a first result, we find that both commodities and real estate exhibit potentially interesting features in an asset-liability management context. Both outperform the liability on average, an exhibit inflation-hedging potential that increases in the long run. In fact, real estate exhibits shortfall probability figures that are as competitive as stocks and that sharply decrease with the investment horizon (from 37% for three years to 15% for 30 years). Commodities substantially outperform bonds in terms of average funding ratio and shortfall probabilities. So we expect significant gains from adding commodities and real estate to the pure liability-hedging portfolio invested in TIPS.

**Liability hedging portfolios**

The results from the previous section suggests that introducing commodities and real estate, in addition to TIPS, in a pension fund LHP would allow for upside potential while limiting shortfall probabilities to a reasonably low level, at least from a long-term perspective. In what follows we quantify the trade-off between a deviation from the perfect liability match and the resulting return upside potential, which in turn will have the welcome side effect of decreasing the required contributions. The consequences in terms of ALM risk budgets of introducing alternative asset classes so as to design enhanced liability-hedging portfolios with improved performance are quantitatively analysed in our full report.

To analyse the characteristics of LHPs that are enhanced by commodities and real estate assets, we proceed in two steps. First, we find the optimal portfolio mix of commodities and real estate and secondly, this portfolio is added to TIPS in various proportions so as to form the enhanced liability hedging portfolio (henceforth enhanced LHP). The first step is addressed by finding the portfolio of commodities and real estate that minimises the tracking error volatility with the TIPS portfolio. Figure 4 shows the resulting portfolios as a function of the investment horizon. As evidenced by the graph, the portfolio is well balanced between the two assets, and the position in commodities increases with the investment horizon.

Table 2 presents funding ratio statistics for the enhanced LHP. Various portfolios ranging from 0% to 50% of alternative investments (AI) with the remainder in TIPS, are studied. The mean funding ratio of the enhanced LHP is a simple linear combination of the individual mean funding ratios and the interpretation is straightforward. In particular, panel A of Table 2 shows that the upside potential is an increasing function of the percentage allocated to the alternative assets within the LHP portfolio.On the other hand, the more the investor allocates to alternative assets, the higher the risk of falling severely short of liabilities is. However, the results suggest great gains when stepping from stand-alone alternative asset classes to alternative investment portfolios in terms of shortfall probabilities. Indeed, at all investment horizons, the shortfall probabilities are significantly lower for various versions on the enhanced LHP than those obtained for the stand alone assets (see Table 1).

For instance, for investment horizons of 10 years, shortfall probabilities are as low as 17%, compared to 27-35% on the stand-alone basis. Note that shortfall probabilities depend solely on the investment horizon and not on the fraction allocated to the AI portfolio. This result is intrinsic to the buy-and-hold methodology and the fact that the TIPS portfolio exhibits non-stochastic funding rations equal to 1.

For 20 year horizons, these probabilities fall to as low as 9% (19-29% on the stand-alone basis). More importantly perhaps, panel C indicates that the decrease in severe shortfall probability is substantial. Even for a substantial allocation to the AI portfolio of 50%, severe shortfall probabilities are only 6% in the short run, and 2% for long investment horizons.

For modest allocations to the AI portfolio (0-15%), the severe shortfall probability even decreases to 0%, meaning that none of the 5000 simulated paths yields a funding ratio lower than 90%, whatever the investment horizon. Overall, these results suggest that the introduction of alternative investment vehicles may lead to increased upside potential for the LHP without severely increasing the shortfall risk.

*This is an extract from the full EDHEC Risk and Asset Management Centre Publication, “Alternative Investments For Institutional Investors: Risk Budgeting Techniques In Asset Management and Asset Liability Management,” sponsored by Morgan Stanley. For a full copy of the report, please contact Séverine Anjubault on France +33(0) 493 187863 or email severine.anjubault@edhec-risk.com *

__ABOUT THE AUTHORS__

**NOËL AMENCProfessor of Finance and Director of Research and Development, EDHEC Business School**

Amenc heads the Risk and Asset Management Research Centre at the EDHEC Business School. He has a Masters degree in Economics and a PhD in Finance. He has conducted active research in the fields of quantitative equity management, portfolio performance analysis, and active asset allocation, resulting in numerous academic and practitioner articles and books. He is Associate Editor of the Journal of Alternative Investments and a member of the scientific advisory council of the AMF (the French financial regulatory authority).

**LIONAL MARTELLINIProfessor of Finance, EDHEC Business School**

Martellini is Director of the EDHEC Risk and Asset Management Research Centre, and has consulted on risk management, alternative investment strategies, and performance benchmarks for various institutional investors, investment banks, and asset management firms, both in Europe and the United States. His research has been published in leading academic and practitioner journals, and he sits on the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. He has co-authored and co-edited reference texts on fixed income management and alternative investment.

**VOLKER ZIEMANNSenior Research Engineer, EDHEC Risk and Asset Management Research Centre**

Ziemann holds a Masters degree in Economics and Statistics and a PhD in Finance. His research focus is on the econometrics of financial markets and optimal allocation decisions involving non-linear pay-offs.

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## Issue 145

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