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Toward LDI Solutions That Actually Work

Michael J. Bazdarich, PhD
Product Specialist/Economist

Executive Summary

  • Our approach to customized LDI solutions analyzes empirical data to determine what maturity ranges a plan should attempt to hedge and which instruments—credit or Treasuries—a plan should devote to such ends.
  • The KRD-matching approach that our competitors typically recommend entails a number of shortcomings which inhibit the real-world performance of those “matched” solutions.
  • Rather than trying to hedge all the curve sensitivities of DB liabilities, our customized solutions focus on a few, especially important maturity segments of DB discount curves.
  • These more selective solutions have been found to deliver both lower tracking error and better funded status performance over time, when examined within historical backtests.
  • Furthermore, the actual implementation of these solutions has performed especially well in recent experience.

SECTION 1: Introduction

You are a defined-benefit (DB) pension plan sponsor who has improved funded status to the point where you want to seriously de-risk via a customized Liability Driven Investing (LDI) solution. What kind of allocation should you adopt? Most of our competitors would advise building a solution portfolio that matches the key-rate durations (KRDs)—or the cash flows—of your assets to those of your liabilities.

Western Asset’s construction of customized LDI solutions differs markedly from that approach. Rather than matching KRDs or cash flows, we focus on minimizing real-world risks of the liabilities while also seeking asset returns that can be expected to keep up with the liabilities over time. These objectives are, of course, universally desirable, and one would think that other types of customized solutions would also deliver on these goals. However, the fact is that they don’t.

KRD- and cashflow-matched solutions suffer from four serious shortcomings. First, fully matching all KRDs or cash flows requires much more than 100% funding or else substantial exposure to derivatives. Second, merely matching KRDs or cash flows still leaves substantial basis risk between assets and liabilities. Third, KRD- or cashflow-matched allocations are likely to result in declining funded status over time. Fourth, to effectively hedge liabilities, a plan must allocate assets both to credit and to Treasuries (USTs), but the KRD-matching process provides no guidance on how to split assets between these two.

In contrast, our customized solutions explicitly hedge the empirical sensitivities of liability returns to yield and spread curve shifts while also providing sufficient returns to sustain funded status. This optimization can be performed without having to resort to leverage via over-funding or derivatives. Because the process focuses on real-world risks, it explicitly determines how best to split assets between credit and USTs.

Because a plan generally cannot fully hedge all KRDs or cash flows, our customized solutions feature exposure to select segments of the yield curve—that is, to select KRDs. Our empirical analysis determines which of these have provided the most crucial empirical source of variation in liability valuation, and the process hedges these dominant risks.

SECTION 2: What the Customizers Usually Don’t Tell You

In a typical competitor’s presentation of a customized LDI solution, you will see the KRDs of a plan’s liabilities, and you will see an asset solution that matches all of these. The inference is that such an allocation is attainable and will effectively eliminate tracking error between assets and liabilities. However, there are critical shortcomings of this approach that a typical presentation fails to mention.

First, you can’t match all those KRDs with merely 100% funding. Each cash flow or KRD bucket within the liability valuation is the equivalent of a zero-coupon corporate bond, with duration equal to its maturity. The couponed corporate bonds actually available have duration much lower than their maturity.

Thus, to hedge the KRD at 10-year maturity, for each $1 of liability valuation, the plan would need about $1.40 of 10-year corporates, since the duration of that bond is only about seven years.1 The same problem exists at each maturity between one and 30 years, and the problem is of even greater magnitude for cash flows past 30 years maturity. As we’ll see in Section 3, for DB liabilities with 14 years duration, full matching of KRDs via corporate bonds would require 153% funding. Cash flow matching would require similar over-funding.

Second, while DB liabilities are typically sensitive to AA yields, there are not sufficient AA bonds available to make an all-AA portfolio investible. Bloomberg Barclays estimates total market value of long AA issuance at $202 billion from 91 issuers, most of which bonds are closely held, not traded. A truly investible solution will require bonds of lower quality, and matching KRDs of AA liabilities with, say, A rated bonds generates substantial basis risk, since real-world spreads between AA and A bonds are constantly changing (cf. Exhibit 1).

Exhibit 1
Yield Spread Between Long A and Long AA Credit
Source: Bloomberg Barclays. As of 30 Apr 17.

Third, while earning an AA yield, DB liabilities do not suffer the downgrades or occasional defaults that AA bonds do, let alone those suffered by lower-rated bonds. So, even apart from the effects of movements in yields and spreads, returns on AA liabilities will exceed those on AA bonds, due to the susceptibility of the latter to periodic “credit events.” In order to keep up with its liabilities, a DB plan must undertake some active management risks, either via a (strategic) allocation to lower-rated bonds and/or via (tactical) active management of the hedge portfolio.2 KRD-matching provides no guidance on how best to achieve this.

Fourth, movements in Treasury yields and spreads are quite distinct, and DB liabilities have relative sensitivities to Treasury yield and spread movements that are different from those of corporate credits and obviously different from those of USTs.3 To properly hedge these sensitivities, a plan needs to allocate both to corporate credits and to USTs, but here too, KRD-matching provides no clue on how to do this.

SECTION 3: Which KRDs to Hedge?

If a 100% funded plan cannot hedge all its KRDs, which should it hedge? This is an empirical question, and at Western Asset we answer it by analyzing empirical data. Our Western Asset LDI Optimizer model uses annual holding period, historical data over 1989–2016 to construct asset portfolios that best hedge clients’ DB liabilities over that sample. Exhibits 2 and 3 show the results of that model’s estimations for a set of DB liabilities that features 14 years duration at current yield levels.4

Exhibit 2
Hedging Options for a 14-Year Duration DB Plan
Source: Bloomberg Barclays, US Treasury, Western Asset
Exhibit 3
Customized Solutions for a 14-Year Duration DB Plan
Source: Bloomberg Barclays, US Treasury, Western Asset

Column (1) shows a portfolio that matches all KRDs of the liabilities using AA corporate bonds, with portfolio weights expressed as percentages of liability valuation. In order to fully hedge all KRDs, a plan would need assets equal in value to 153% of the liability valuation. Even then, that allocation would still experience 350 basis points (bps) per year tracking error against liabilities over the sample.

If, instead, the asset portfolio were confined to 100% of liability value, and portfolio weights were chosen to match the weight of each maturity bucket in the liability valuation, the portfolio would look as in column (2). Not surprisingly, this portfolio exhibits much higher tracking error (510 bps) over the sample period, and asset returns underperform liabilities on average by 240 bps per year.

Column (3) shows the portfolio that optimally allocates to the various AA maturity buckets when portfolio value is limited to 100% of liabilities. That portfolio has all assets invested in the longest maturity bucket. Notice that the tracking error result for (3) is substantially smaller than that for (1). While the 15–30 year bucket accounts for only about 68% of the dollar duration of the liabilities, it has accounted for enough real-world volatility in the liabilities that a 100% funded allocation is best off putting all its funds in that bucket. In effect, a KRD-match under-hedges this bucket.

Notice finally that while (3) delivers the lowest tracking error of the alternatives discussed so far, it still does not provide enough return to keep up with the liabilities on average. This should not be surprising, since as mentioned in Section 2, even a perfectly-matched AA portfolio (one of AA zero-coupon bonds, if they existed) would underperform AA liabilities, because of the susceptibility of AA bonds to credit events.

In order to construct a portfolio that has a hope of keeping up with the liabilities, one must introduce higher-yielding, lower-quality bonds into the mix. Given the limited supply of liquid long AA bonds in today’s market, the lower-quality bonds also provide a more “investable” set of allocations.

Columns (4) through (6) describe hedges constructed the same way as those in (1) through (3), but utilizing A rated (or better) bonds in the indices. Here, a KRD-matched portfolio again requires vast over-funding and generates even larger tracking error, now 530 bps per year, as seen in (4). A valuation-weighted portfolio performs just as poorly as seen in column (5). Choosing weights optimally across buckets, as in column (6), again results in all assets allocated to the longest maturity bucket. The tracking error for (6) beats the “investible” KRD match (4) and is as low as for the “uninvestible” KRD match (1).

The allocation in (6) is as good as a plan can get with an all-credit portfolio. Even this allocation underperforms the liabilities on average over 1989–2016.

As mentioned in Section 2, to hedge all the risks DB liabilities exhibit, a plan must allocate to both USTs and credit. The portfolios illustrated in Exhibit 3 do so. Exhibit 3 focuses on A or better corporate bonds, in line with the investibility issues already discussed.

The portfolio in column (7) minimizes tracking error without regard to returns. That allocation invests in only three of the 10 available maturity/quality buckets, but is able to reduce tracking error to 250 bps per year. However, it underperforms liabilities on average over the sample.

Column (8) shows the portfolio that minimizes tracking error while still at least matching liability returns on average. This portfolio also allocates to only three of the 10 buckets, but still is able to reduce tracking error to 350 bps per year, the same as allocation (1), even though it requires considerably fewer assets and is actually investible.

The portfolio described in (8) is as good as it gets against these liabilities for an all-physical-asset allocation. Usage of leverage via swaps or futures would allow tracking error to be reduced somewhat further, but otherwise such solutions would look similar—invested in only a few maturity buckets. We leave the details of custom allocation with derivatives for another time.

The results here are specific to the 14-year duration liabilities used, but solutions for other cash flow sets are similar. An appendix5 to this paper reports results for a cashflow stream with only 11 years duration. Tracking error is lower across the board for these solutions, and, naturally, the optimal allocations are more heavily allocated to intermediate maturity buckets. However, for these solutions as well, investments are required in just a few of the available buckets, and such optimized allocations outperform KRD-matched solutions, despite the huge over-funding of the latter. Also for these liabilities, some allocation both to USTs and STRIPS is necessary for a fully effective hedge.

The appendix5 also reports results of hedging exercises using the Treasury’s High Quality Market (HQM) yield curve (A or better) to evaluate liabilities. Such solutions feature smaller allocations to STRIPS and lower tracking error levels than those reported in Exhibits 2 and 3, but otherwise the results are comparable.

SECTION 4: Case Study of an Actual Custom Solution

The hedging exercises described above concerned liabilities evaluated with generic yield curves based on all available corporate bonds and a transparent curve-generating process. We can estimate such curves very far back in time and construct a sample period that spans a wide range of historical experience.

However, some clients we work with utilized stylized discount curves produced by their actuaries. The processes underlying these curves are proprietary, and only short historical samples of these curves are available. When asked to construct customized solutions to track such liabilities, we perform a slightly different process from that of Section 3. The sample period is shorter and less varied, and we typically optimize using monthly holding periods for liabilities and assets. This section describes the hedge construction process utilized for one such client we work with.

For this client, the history of the yield curves utilized extends back to January 2012, and as of 12/31/16, said client’s cash flows exhibit a duration of 11.5 years. We utilize a set of constant-maturity A or better corporate bond indices constructed from the Barclays Point universe, along with 25-year STRIPS.

In Exhibit 4, the allocation in (9) is that which produces the lowest tracking error (31 bps per month, or 106 bps per year), while also matching or beating liability returns on average over the sample. Our investment team actively manages a portfolio that utilizes the allocation in (9) as a custom benchmark. In this way, we attempt to manage directly against the client’s liabilities.

Each month, we redo the optimization analysis—using a now-expanded sample with an extra month of data—to determine if a new allocation, based on all available data, could meaningfully outperform the old custom benchmark (constructed over the 1/12–12/16 sample). This comparison provides an indication of how well our custom benchmark is performing.

Column (10) shows how the custom benchmark is performing over the now longer sample. Column (11) shows the allocation that minimizes tracking error—without regard to total return—over the new sample. Column (12) shows the solution that minimizes tracking error over the extended sample while still at least matching the average return of the liabilities.

Exhibit 4
Custom Benchmark Construction
Source: Bloomberg Barclays, US Treasury, Western Asset

As seen in Exhibit 4, allocations (11) and (12) are little different from our custom benchmark (10), and their performances across the extended sample—both average excess returns and tracking errors—are also little different. Furthermore, for each month of 2017 to date, the monthly misses between custom benchmark and liability returns have been very small, well within the 31 bps per month tracking error band shown in Exhibit 5, as estimated by our initial optimization run.

Exhibit 5
Monthly Performance of Custom Benchmark Versus Liabilities
Source: Bloomberg Barclays, US Treasury, Western Asset. As of 30 Apr 17.

All of these results testify to a satisfactory performance of the custom benchmark so far this year. Were the observed misses larger or if the re-run optimization results pointed to a markedly different allocation, the inference would be that current experience were deviating from the historical sample and that some recalibration of the custom benchmark were in order. This has not been the case so far this year.

When actuarial projections are re-estimated each year, a full optimization run is again performed. That process helps recalibrate the custom benchmark and the actual portfolio, if such changes are called for.

Finally, note that the annualized tracking error estimates shown in Exhibit 4 are markedly lower than those reported in Exhibits 2 and 3. This is partly due to the lower duration of the liabilities in play here (11.4 years versus 14.9 years), but it is mostly due to the natures of the two sample periods. The 1989-2016 sample period analyzed in Section 3 includes three financial crises. In contrast, the 2012-16 sample period analyzed here has been a relatively tranquil episode. Hopefully, the custom benchmark described here will still perform well against these liabilities during a period of financial stress, but it is only to be expected that monthly and annual misses would be larger in that event.

SECTION 5: Solutions Should Work in the Real World, Not Merely on Paper

Derisking via a customized LDI solution should simplify the pension management process. This will be the case only if the customized solution actually works, providing minimal tracking error and sustaining funded status over time. KRD- or cash-flow-matched solutions work great on paper, but they don’t work so well in practice.

In the real world, credit spreads are constantly changing, both across maturities and across quality ratings. Merely matching KRDs or cash flows fails to allow for movements in credit spreads. Similarly, credit events do occur. Not only do such events induce further tracking error between assets and liabilities, but also they can cause declining funded status over time. Finally, in addition to these pitfalls, full KRD- or cash-flow-matching requires vastly more physical assets—or more exposure to derivatives—than purveyors of such solutions are wont to admit.

The solutions we construct, as described in this paper, are designed to address these real-world pitfalls head on. Our process attempts to identify the dominant risks a DB plan will face, in order to efficiently hedge these risks without having to resort to derivatives or over-funding. Because our solutions are calibrated to historical experience, we can determine how best to meld credit and USTs and how best to craft a solution that has a chance of sustaining or improving funded status over time.

It might be said that these solutions assume past behavior will recur in the future. It is true that if relative performances of yield or spread curves were markedly different from historical experience, the hedges constructed here would not perform as well as we depict. However, solutions based on KRD-matching assume that corporate yield spreads do not change at all and that credit events don’t occur, since only in such an environment would KRD-matched solutions effectively track DB liabilities.

Our solutions can be seen to be optimal over well-defined historical samples. KRD-matched solutions are not optimal over any historical time period, nor will they be in any future time period, because credit spreads will inevitably vary, both across maturities and across quality levels, and downgrades do occur.

Our analysis has shown that solution portfolios constructed from a few key maturity “buckets” can track DB liabilities more closely than key-rate-duration-matched solutions, even while avoiding the over-funding or leverage that “matched” solutions require. We are comfortable with the reliance of our methods on historical samples, and our regular monitoring of these constructions provides real-time information as to when our solutions should be rebalanced.


  1. And the corporate bond with 10 years duration has about 14 years maturity, thus substantial basis risk against the 10-year yield.
  2. For a detailed discussion of this issue, See Effective LDI: Keeping Up With Your Liabilities, August 2013, available on our website.

  3. See A Risk Factor Based LDI Analysis: A 100% Hedge Ratio Is Not Enough May 2015, available on our website.
  4. Previous papers utilized regression analysis to pursue much these same questions. See LDI and the Persistence of Tracking Error, January 2013 and Persistence of Tracking Error Part 2: Corporate Bonds As LDI Hedges, October 2013, both available on this website. The advantage of utilizing the Optimizer model for these questions is that we can impose constraints on portfolios that are not easily replicable within regression analyses, namely that portfolio weights not be negative and that average returns on assets and liabilities be equal.

    The Optimizer model generates a series of liability returns by shocking a client’s cash flow obligations with movements in a DB pension discount curve. The curves used within our Optimizer model were described in detail in the papers referenced here. These curves were derived from AA Corporate bond data within the Barclays POINT universe. Spot yield curves were fitted using the cubic spline technique created by the US Treasury for formulation of its HQM corporate yield curve (established by the Pension Protection Act). Against this sample of liability returns, historical return data for various assets are utilized to determine effective hedge portfolios. For the corporate bond indices used in the text, bonds were chosen from the same universe as was used to construct the pension discount curves, and maturity buckets were chosen to match the spline segments of the pension curves, so as to provide the best possible hedge of liability return so generated.
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