Active Liability-Driven Investing and Pension Management

Western Asset

Executive Summary

  • Portfolio management under a “surplus optimization” framework prescribes very different optimal practices from those holding under a “total-return” framework. While cash is a low-risk asset with respect to total returns, it is high-risk relative to surplus. Long bonds are a high-risk asset with respect to total returns, but they can be a low-risk—even minimum-risk—asset with respect to surplus. In either framework, aggressive allocations should be concentrated in assets exhibiting favorable risk/return trade-offs relative to cash and other assets.
  • Liability-Driven Investing (LDI) has been incorrectly identified with passive, fully-hedged asset allocations that mimic the interest rate sensitivity of liabilities. True LDI principles—what we call active LDI— use surplus optimization techniques to enhance portfolio return and/or to hedge non-interest rate risks of liabilities, in addition to addressing interest rate risks.
  • A “fully-hedged” or dedicated bond allocation passes up other assets with extremely favorable return/risk trade-offs. Such a position would be taken only by a plan with no risk tolerance.
  • Active LDI strategies may be economically relevant for a defined benefit pension plan. They are certainly relevant for compliance with pension accounting and regulatory institutions.

I. Introduction

This paper analyzes the principles of liability-driven investing: LDI. Under an LDI investing framework, asset allocations are specifically targeted at achieving or servicing a particular liability target. In effect, the liabilities become the benchmark for the investment process, and its success is judged solely in terms of whether or how closely investment returns come to achieving the liability benchmark. In contrast, asset allocation has traditionally been done within what we will call a “total return framework” in which the success of the allocation is judged solely in terms of whether total returns are high or not.

Now, any investment is made with some goals in mind. Within a total return framework, the goals of the investment process are peripheral to investment decisions. They may affect the “benchmark” holding period, the degree of risk budgeted, or the asset benchmarks selected, but they do not explicitly enter the portfolio selection process. Within an LDI framework, the goals of the investment process—the liability target—are central to this process; again, they are the benchmark for it.

As we have defined it so far, LDI is still an amorphous concept. Saying that investment decisions are benchmarked to liabilities doesn’t tell us much until we know what drives movements in liabilities. For example, if streams of future obligations are evaluated at a discount rate that is the same as the rate of return on assets, then liabilities will automatically vary with assets, and one allocation will “serve” liabilities as well as any other.

For private sector pension plans, both accounting and regulatory standards here and abroad mandate a bond-equivalent valuation of pension liabilities. Future liabilities are evaluated with a discount rate equivalent to market yields on long-term bonds. As a result, liability valuations are highly sensitive to market fluctuations in interest rates. Therefore, an LDI framework under these conditions results in very different investment practices from those that have evolved under the total return tradition.

Within a total return framework, cash is a low-risk asset. Holding-period returns on long bonds fluctuate sharply from year to year, and with the majority of fixed-income securities in short to intermediate maturities or durations, long bond returns vary substantially relative to fixed-income market benchmarks. On both counts, long bonds are high risk.

The situation is completely different within an LDI framework (that utilizes bond-equivalent valuations of liabilities). Here, cash is a high-risk asset, since its low duration provides little or no correlation with bond yield-driven swings in liability valuations. Similarly, long bonds are a low-risk asset here, because their swings in valuations cohere closely with those of liability valuations.

In practice, LDI has come to be associated or identified with “fully-hedged” asset allocations: investments in a “dedicated bond portfolio” which has an income stream matching the payment flow for liabilities. However, this is an extremely narrow depiction of LDI principles. Just as one can benchmark an equity allocation to the S&P 500 without investing in an index fund, so too one can invest according to a liability benchmark without matching or “immunizing” relative to that benchmark.

We will describe here a range of “active LDI strategies:” strategies that operate within an LDI framework without necessarily hedging away all interest rate risk (just as an equity allocation that benchmarks to the S&P 500 but does not commit to an indexed fund is called an active equity allocation). We will show that a fully-hedged (passive) LDI strategy should rarely be undertaken, because it leaves too favorable a return/risk trade-off unexploited. We will also describe some active LDI strategies that seek to hedge risks other than interest rate ones and which, as a result, can provide a better hedge of all liability risk than an immunized allocation.

Section II formally contrasts a total return optimization process with an LDI or “surplus optimization” process. Section III draws on these results to describe active LDI strategies. Section IV summarizes our results, puts them into an historical context, and sets the stages for further analysis. There is a range of conceptual issues underlying our analysis. Rather than presenting an exhaustive discussion of them, we will touch on additional points in a more extensive set of footnotes than we would usually provide.

II. Total Return Optimization versus Surplus Optimization

We’ve already stated that portfolio optimization under an LDI framework provides extremely different results from those holding within a total return framework. We will outline those differences here. We choose a 1960-2005 sample period for asset returns, and we’ll see that this period is not a particularly favorable one for long bond performance. Nevertheless, even over this sample period, long bonds are the dominant asset for portfolios chosen within an LDI framework.

Under what we have described as a total return framework, portfolio optimization amounts to maximizing expected total return for a given level of total return risk. Under an LDI framework, portfolio optimization amounts to optimizing surplus. Surplus is defined as assets less liabilities: essentially the “alpha” of the portfolio relative to the liability benchmark. The higher the return on assets relative to that on liabilities (the higher the alpha) the better is performance. Similarly, the lower the variability of assets relative to liabilities (the lower the tracking error), the better is performance. Surplus optimization amounts to maximizing surplus return for a given level of surplus risk.1

For a total return framework, Exhibit 1 lists expected returns and risks for an asset universe of cash equivalents (1-month Treasury bills), intermediate Treasury notes, long-term corporate bonds, and large cap stocks as reported by Ibbotson and Associates over the 1960-2005 sample and annual holding period returns. Exhibit 2 depicts the frontiers of efficient portfolios in that framework: those portfolios achieving maximum expected return for a given level of risk, one frontier utilizing swap/overlay positions, the other not.2

Exhibit 1
Total Return Characteristics (%/yr.), 1960-2005
Source: Ibbotson Associates, Western Asset

Again, this is not a particularly favorable sample period for long bonds. Their mean return barely exceeds that of intermediate notes, the Sharpe ratio for bonds is lower than those for notes and equities, and bonds lie far off the efficient frontiers. In fact, long bonds receive zero portfolio shares over most of the no-swaps frontier. Over the with-swaps frontier, most of the efficient portfolios include short positions in long bond swaps. (We comment below on the advisability of swaps and leverage in these and in LDI portfolios.) Similarly, as expected, cash equivalents are the lowest-risk asset and lie virtually on both frontiers. Cash receives near-100% portfolio weights in the lowest-risk segments of the frontiers (i.e. around the “Cash” point).

Exhibit 2
Efficient Frontier for Total Return Optimization
Source: Ibbotson Associates, Western Asset

Observe how things change when we shift to a liability driven, surplus optimization framework. Exhibit 3 shows risk-return characteristics for the same asset set, sample period, and holding period as before, but now with risks and returns calculated relative to a liability mandate. Specifically, we assume that liability valuations vary with long bond rates, as well as with a 3% per year rate of increase—to cover expected wage inflation—and additional variability (noise) to simulate the effects of unexpected swings in inflation and actuarial factors.4 Exhibit 4 shows the efficient frontiers for a surplus optimization framework.

Exhibit 3
Surplus Return Characteristics, 1960-2005
Source: Ibbotson Associates, Western Asset

As stated earlier, in this context, cash is now the highrisk asset, while long bonds are the low-risk asset. Long bonds dominate cash and notes, off ering both higher expected returns and lower risk with respect to liabilities. Not surprisingly, then, long bonds lie right on the effi cient frontiers for surplus optimization, while both cash equivalents and intermediate notes are well off the frontiers.

Exhibit 4
Efficient Frontier for Surplus Optimization
Source: Ibbotson Associates, Western Asset

Similarly, long bonds account for all the fixed-income exposure in optimal portfolios on the “no-swaps” efficient frontier in Exhibit 4. The optimal surplus allocation decision is merely a question of how much to allocate to long bonds, in order to reduce surplus risk, and how much to allocate to equities, in order to achieve more favorable surplus returns.5

Notice that even over a sample period where long bonds are an inferior asset in terms of total return, they are nevertheless the dominant asset in terms of surplus. In a total return framework, cash is a perfectly acceptable spot to “park” new or otherwise uncommitted funds, because of its low risk, and long bonds should be utilized only as a vehicle to actively “bet” on falling interest rates. However, in a surplus optimization framework, cash is high-risk. New or uncommitted funds should be parked in long bonds, and cash should be held only to actively bet on rising interest rates.

With bonds so closely correlated to liabilities, one might guess that any leverage in an LDI framework would also be taken in long bonds, but this is not necessarily the case. Before discussing this point, though, it is worthwhile to confront the issue of leverage, specifically swaps, head on. Many investors and advisors would argue that leverage provides too much risk to have a place in a pension allocation or, perhaps, in any asset allocation. This view misses several important perspectives.

A pension allocation is already leveraged. The pension plan owes its liabilities to its beneficiaries, and so its investment operations involve the utilization of “borrowed funds,” the very definition of leverage.6 Investing in equities within a pension fund is exactly the same thing as shorting bonds in order to go long equities, and yet equity investments are commonly found in pension plans. The “leveraged” positions that the optimization process here prescribes involve long and short positions in various fixed-income assets. While this is indeed leverage, it is subject to a much different (lower) order of risk from those one undergoes when shorting cash and going long equities, commodities, or real estate.

Furthermore, the leveraged positions prescribed here can be seen to reduce risk relative to non-leveraged positions. Leveraging a high-Sharpe-ratio asset is a less risky way to achieve a given level of return than a non-leveraged position in a low-Sharpe-ratio asset.7 Thus, the “with-swaps” frontiers in both Exhibits 2 and 4 lie to the left of the no-swaps frontiers: they achieve the same level of expected return with less risk, with much less risk in the case of surplus optimization.

We are not advocating the usage of leverage in pension allocations, we are only detailing its possible advantages and pointing out that many of the dicta against leverage are inappropriate for this case.

It turns out that the practice of taking leverage in high-Sharpe-ratio assets is just as relevant in a surplus framework as it is in a total return framework. While bonds are the low-risk asset in a surplus optimization framework, leverage involves borrowing against cash and other assets, and so the same factors which determine optimal leveraging in a total return framework are also applicable in a surplus framework 8 In other words, the optimal portfolios on the with-swap frontiers in both total return and surplus frameworks (Exhibits 2 and 4) are dominated by leveraged positions in notes, which, in the present sample, exhibit the most favorable Sharpe ratios.

This can be seen by comparing the portfolio compositions in points A and B in Exhibit 2 and points C and D in Exhibit 4. Point A describes a leveraged allocation that optimally replicates the total return on an all-equities portfolio within a total return framework; point C describes an allocation that optimally replicates the return on equities with leverage within a surplus framework. Point B provides the same total return risk as point A, but without leverage, while point D provides the same surplus risk as point C, but without leverage. The portfolio compositions of these points are shown in Exhibit 5.

Exhibit 5
Representative Portfolios in Total Return and Surplus Frameworks
Source: Ibbotson Associates, Western Asset

In both frameworks, allowing leverage and holding expected returns constant results in lower equity allocations, allocations to note swaps, and lower overall risk than in an all-equities allocation. Moving from point A to B and from C to D (that is, keeping risk constant but disallowing leverage) results in an equity allocation that is about unchanged in each case. In the total return framework, the “cash” position in notes is also about unchanged. In the surplus framework, the exposure in notes is transferred into long bonds when leverage is disallowed.

Other points on the frontiers are similar to those described here. Within the total return framework, as risks and returns rise, without swaps, note positions decline and equities increase, while with swaps, note swap positions rise and equity allocations rise slightly. Within the surplus framework, as risks and returns rise, without swaps, bonds positions decline and equities increase, while with swaps, note swap positions rise alongside moderate increases in equity allocations. Exhibit 6 summarizes the various results derived in this section.

Exhibit 6
Summary of Results for Optimal Portfolio Management
Source: Western Asset

III. Active LDI

As we have seen, an LDI approach invests according to a liability benchmark, and with a bond-equivalent valuation of liabilities, an LDI framework yields the surplus optimization results derived in the preceding section. Now, LDI has become commonly associated with immunized or dedicated bond allocations: attempts to attain a point of minimum surplus risk (the point furthest to the left on the frontiers in Exhibit 4). However, as this immunized allocation comprises but a single point along a continuum of acceptable surplus-optimized allocations, it is clearly a narrow interpretation of LDI principles.

If liability valuations were sensitive only to swings in interest rates, a minimum-risk (fully-hedged) allocation would become a no-risk allocation, and it would make sense to pursue it. In reality, even a “fully-hedged” allocation leaves the pension plan susceptible to risks from changes in actuarial assumptions and in workers’ wage levels. Once any non-interest risks are acknowledged for liabilities, even a “fully-hedged” allocation exhibits positive risk, and, more importantly, the efficient frontier exhibits vertical slope at the minimum-risk point, as indicated in Exhibit 4.10

That is, at the fully-hedged position, there is an infinitely-favorable trade-off between return and risk. The plan can achieve very large incremental increases in expected surplus return in exchange for incurring tiny incremental risks by moving away from a fully-hedged position. If the plan sponsor has any tolerance for risk, it would not choose to operate at a fully-hedged or immunized position.11 So not only is a (passive), fully-hedged allocation a special case of LDI, it is a largely irrelevant special case.

Active LDI strategies involve taking active asset positions in order to exploit the very favorable risk/reward trade-offs that are available upon moving away from passive, fully-hedged allocations. In a no-leverage situation, this involves adding exposure to equities or to other asset classes not included in the Section II discussion. For example, an active long-duration allocation, managed according to a long-bond benchmark (with duration equal to that of liabilities), would achieve nearly all of the “hedging” advantages of an immunized portfolio, but it would also leave open the chance for positive returns on surplus (alpha).12

With the ability to utilize leverage, even the limited set of assets considered in Section II can deliver a wide range of aggressive allocations. The leveraged surplus portfolio (point D from Exhibit 4) discussed above is an example. A long position in note swaps can provide the same aggregate duration as a non-leveraged position in long bonds. Therefore, it can provide an approximate hedge of liabilities’ interest-rate risks, while also providing, in the right circumstances, higher returns. Such a “duration hedge” would leave the plan susceptible to changes in the slope or convexity of the yield curve. However, that could be an effective trade-off for the higher return it offers or in particular market conditions. That is, a leveraged note position would not be suitable purely as a hedge of liabilities, but it could be effective as an active strategy aimed at achieving both higher returns on surplus and minimal incremental surplus risk.

Usage of derivatives leaves the allocation subject to margin risks. Though adverse movements in interest rates (higher bond yields) should generate offsetting gains and losses, the “gains” on liability valuations are not “bankable,” while the losses on fixed-income swaps necessitate margin maintenance. However, as long as the aggregate duration of assets and liabilities are roughly equal, interest rate swings should not generate substantial swings in surplus, and so the portfolio should automatically have sufficient assets available to meet the margin requirements for the leveraged positions.

For more aggressive allocations, this might not be the case. Even there, keep in mind that the associated risks are clearly less than those for an equally aggressive, non-leveraged allocation. So any heartache associated with momentary margin calls on a swap position will be much smaller in magnitude and less frequent than the heartaches from cash losses occurring more frequently and substantively in an equally aggressive, non-leveraged allocation (in equities).

So far, we have analyzed active LDI as a return-enhancement tool. It could be that a dedicated bond allocation would not be a minimum-risk allocation. That is, we mentioned above that liability valuations are subject to non-interest risks. If certain assets were found to be correlated with actuarial or inflation risks, allocations to these could reduce surplus risks below those on a dedicated bond allocation. For example, a bond swap overlaid onto a long position in TIPS or commodity futures/swaps overlaid onto a long bond allocation could address both interest and inflation risks.13 Similarly, if it could be determined that other assets correlated with the actuarial risks of the liabilities, positions in those assets could also be overlaid onto the allocation. If swaps or futures were available in all the relevant assets, these could all be put on, with collateral allocated elsewhere to generate “alpha” (surplus returns).

The exact form of such allocations that would be practicable for a specific pension plan depends on the time-pattern and risks of the plan’s future obligations, so it is impossible to present one empirical example that would be informative for all plans. Suffice it to say that there are a multitude of asset and asset-overlay strategies that are available to a pension plan when it is operating within an LDI framework. Some of these allocations will be aimed at exploiting favorable risk/reward trade-offs. Others might be aimed at improving on the hedge provided by a dedicated bond portfolio. However, each of these alone or in tandem are valid examples of liability-driven investing and are dramatically different from the passive, dedicated bond allocations that are commonly identified with LDI.

IV. Conclusions

We have seen that the switch from total-return to surplus optimization results in some dramatic shifts in prescriptions for optimal asset allocation, especially for non-leveraged positions. Cash is a low-risk asset in a total return framework, but it is a very risky one in a surplus framework. Bonds may be risky in a total return framework, but they are low risk in a surplus framework. Beyond that, we have found that even within a surplus optimization framework, any leverage should be taken in assets exhibiting favorable return/risk trade-offs and not necessarily in those assets that correlate highly with liabilities.

We analyzed a liabilities valuation with interest-sensitivity (duration) essentially identical to that of long-term corporate bonds. With other liability “benchmarks,” long corporate bonds might not provide as close a hedge as they did in our example, but there will always be some fixed-income instrument with duration and interest-sensitivity very close of that of liabilities, and the results derived here for “long bonds” will hold for that instrument.

While LDI strategies are often taken as synonymous with dedicated, passive bond allocations, we have shown that this is a special case of LDI and a low relevance one at that. Investors with any appetite for risk and also those looking for more effective hedges of all the risks associated with their liability targets will find more effective allocations within the range of active LDI allocations, making use of equities, active fixed-income, and, perhaps, overlays and derivatives. Still, these tools will be applied differently within an active LDI strategy from the way they would be utilized within a total return framework.

Now, we argued in a preceding report that the reforms pending in pension regulatory and accounting procedures may not have much effect on pension investment operations.14 It may be that the findings on surplus management elaborated here will have more impact than the forthcoming reforms.

The problem with this statement is that all the strategies we have elaborated here have been relevant for twenty years, ever since the inception of FAS 87. The definition of surplus analyzed in Section II—based on a bond-equivalent liability valuation—was introduced by FAS 87, and the LDI results we have presented were anticipated as early as 1988, once FAS 87 was in place.15 Furthermore, we emphasized previously that many empirical studies have found that financial markets already fully reflect pension funded status and riskiness in pension sponsors’ market valuations. This implies that firms should already be working to optimize pension plan surplus performance if only to manage the impact on their market valuations.

Despite these results, the fact remains that many pension funds continue to operate explicitly or implicitly within a total return framework. Merton Miller once remarked that “a disequilibrium that has lasted for 30 years and shows no signs of abating is too hard for any economist to accept.”16 Economists have been arguing for thirty years that pension funds should pursue an all-bonds allocation. For almost twenty years, they have been arguing that surplus optimization results in different asset allocations from those that make sense under total return optimization. With total return tactics still prevalent in the face of these theoretical prescriptions, one must at least question the real-world relevance of the bond-equivalent valuations of pension liabilities that gave rise to our Section II analysis.17

This raises the question of just what is the right way to manage a pension fund? We won’t try to resolve this question here. If you are content with a total return mandate for your pension fund, that is fine. We would only point out that the surplus valuation framework analyzed here is consistent with the constructs embedded in current FASB standards and ERISA regulations, and it will become even more relevant for accounting and regulatory compliance purposes when impending reforms are in place. If these institutional structures have any relevance for you or if you accept the bond-equivalent valuation of liabilities as being economically relevant for your plan, then the analysis pursued here is relevant as well, and you might benefit from consideration of the results we have presented. Assuming that is the case, our next report will analyze how the results and strategies discussed here differ depending on whether a plan is under- or over-funded and whether it is young or mature.


Arnott, Robert and Peter Bernstein (1988), “The Right Way to Manage Your Pension Fund,” Harvard Business Review, Jan./Feb., P. 95-103.

Miller, Merton (1977), “Debt and Taxes,” Journal of Finance, P. 261-273.

Western Asset Management (2006a), “Which Alpha Would You Choose?”.

Western Asset Management (2006b), “What U.S. Pension Reform Will Mean,”.

  1. We have intentionally described LDI and surplus optimization here to parallel the portable alpha strategies described in Western Asset (2006a). Meanwhile, although an investor must optimize over a lifetime horizon, it is well understood that a lifetime optimization process has embedded within it a one-period optimization “problem” analyzed here in terms of “efficient portfolios.” Also, the “holding period” studied in a “one-period” analysis can be made as long or as short as is most relevant for the decision-making process.
  2. The return-maximization process can be taken with or without a constraint that all portfolio weights are non-negative. If all weights are non-negative, then no asset can be shorted. If negative weights are allowed for some assets—with the sum of all weights still equaling 100%—then the process may allocate negative weights to some assets. Strictly speaking, negative weights would be interpreted as short positions. In the real-world, such short positions could best be accomplished by swaps, as, say, a long-bond swap, is identical to a long position in long bonds and an offsetting short position in cash equivalents. In this analysis, we interpret short positions-when they arise—as being equivalent to positions in swaps. Note also that upon removing the non-negativity constraint, the resulting (with-swaps or with-shorting) efficient frontier will always lie outside (to the left of) the constrained (no-shorting) frontier. That is, without the no-shorting constraint, expected returns for a given level of risk will be greater, and risks for a given level of return will be lower.
  3. Narrowly defined, the Sharpe ratio is the return/risk ratio for an asset measured relative to the risk-free asset. One point we emphasize in this paper is that there is no such thing as a risk-free asset. To measure something comparable to the Sharpe ratio in this context, we measure the ratio of return to risk, when these are calculated for each asset relative to cash, e.g., the ratio of excess return of bonds over cash divided by the standard deviation of bonds less cash. To calculate this, one must know the covariance of bonds and cash, as well as the items listed.
  4. The fact that liabilities have an expected return equal to bond yields plus an expected wage inflation factor of 3% per year makes them equivalent to an accumulated benefit obligation (ABO) concept, as described in Western Asset (2006b). Benefits are ultimately based on salaries as of retirement or termination, but ABO valuations measure benefits based on current salary levels. Therefore, ABO valuations will rise over time both with interest rates—as future obligations move closer in time—and with wage increases. PBO valuations are based on actuarial assumptions of terminal salary levels, so they rise over time with interest rates and only with unexpected changes in wages: wage changes different from actuarial assumptions. Both ABO and PBO valuations vary over time with changes in actuarial assumptions, so the random noise factor in our liability valuation can be construed as proxying for both actuarial changes and unexpected inflation. This liabilities representation could also be considered equivalent to a PBO valuation for a plan sufficiently under-funded that the “return” on liabilities would be expected to be 3% per year greater than that on an all-bonds allocation of existing assets.

    Keep in mind that the qualitative results derived here for optimal portfolios are independent of the absolute level of surplus returns utilized. Choosing a diff erent expected return on liabilities shifts the whole surplus-optimization effi cient frontier vertically, but it does not alter portfolio composition on the frontier. Choosing a diff erent standard deviation for liabilities—but leaving covariances with assets unchanged—shifts the frontier horizontally, leaving portfolio compositions unchanged along the frontier.
  5. All-bond allocations provide negative expected return on surplus for reasons discussed in footnote 4.
  6. One would not typically lend at the bond rate in order to invest at cash yields, and this is exactly why the optimal portfolios in surplus optimization space do not feature positive cash positions. Also, a fully-hedged or dedicated bond allocation is the only pension allocation that involves no leverage.
  7. Western Asset (2006a) makes this same point with respect to alpha sources and information ratios.
  8. That is, an asset’s return, risk, and covariance against other assets are what determine its efficiency as a leverage vehicle in both frameworks. More than Sharpe ratios are involved here. Otherwise, in the total return framework, leverage would have been taken in equities, since they exhibit the highest Sharpe ratio in Exhibit 1. It is a standard finding that in a portfolio setting, what matters for an asset is not its risk per se, but its riskiness relative to the portfolio: its “beta.” In the same way, in the optimization processes analyzed here, leverage is taken in assets whose returns relative to the portfolio return (excess returns) are favorable fractions of their beta relatives to the portfolio. As such “favorable” assets then become more prominent in the portfolio, their betas rise and those of other assets fall. The optimal portfolio is achieved when the ratio of “excess return” to beta is the same for every asset. The point is that in both total return and surplus frameworks, leverage is optimally taken in assets with (initially) high ratios of returns relative to their betas. Our reference to high Sharpe ratios is a euphemism for this feature.
  9. Keep in mind that the “returns” and “risks” shown side by side for the frameworks are different concepts, with the total return framework reporting portfolios’ total returns and risks on total returns, while the surplus framework reports portfolios’ surplus returns and surplus risks.
  10. If liabilities were subject only to interest-rate risk, then, again, bonds would be a zero-risk asset, and so the efficient frontier would be linear, with a kink at the minimum risk point. However, with non-interest riskiness for liabilities, the frontier becomes curved and smoothly so. Therefore, at the point of minimum-risk, the frontier must have zero slope in risk/return space, which means that it must have infinite—vertical—slope in the return/risk space mapped in Exhibit 4.
  11. Notice that this is true regardless of whether or not leverage is allowed. Both the no-swaps and with-swaps frontiers have vertical slopes at the minimum-risk position in Exhibit 4. It is true that the with-swaps frontier loses steepness more slowly than does the no-swaps frontier, so that an investor with a given level of risk tolerance would move farther away from full hedging under a with-swaps mandate than he would under a no-swaps mandate. Still, both curves are vertical at the minimum-risk point, so that some incremental risks would be undertaken by any investor regardless of whether or not leveraging were allowed.
  12. Seeking “alpha” in other classes relative to the liability benchmark would be equivalent to portable alpha strategies with a beta position in the liabilities. Western Asset (2006a) discusses portable alpha strategies relative to a large-cap equity beta. Similar principles apply here.
  13. Such a position would also be superior to a dedicated TIPS portfolio, if one existed. A dedicated TIPS portfolio would hedge the inflation risks of the liabilities, but it wouldn’t hedge the sensitivity of liabilities to swings in nominal interest rates.
  14. See Western Asset (2006b).
  15. See Arnott and Bernstein (1988). That paper laid out how relative risks among assets and the associated efficient frontier for investing shift when moving from a total return to surplus framework. We have provided some specific results that they didn’t. For example, their analysis does not make clear that fully-hedged asset allocations leave extremely favorable risk/reward trade-offs unexploited, nor do they point out that hedged allocations often deliver negative expected surplus returns. Still, our analysis is little more than an application of their results.

  16. Cf. Miller (1977), p. 266.
  17. Again, our application of LDI in Section II proceeded from a bond-equivalent valuation of liabilities, which made liabilities highly sensitive to interest rate fluctuations and thus made long bonds the low-risk asset. If one were to reject this notion and postulate that “true” liability valuations were sensitive to other factors, an LDI analysis could still be pursued, but the correlations between assets and liabilities would be completely different, so that the resulting efficient frontier for surplus optimization would be completely different as well. We are not advocating that this be done, but merely pointing out that our more general depiction of LDI practices is itself a special case of an even more general LDI framework within which liability valuations might be sensitive to any number of factors.
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