• Utilizing an actual set of client liability cash flows and their evolution over time, we identify the customized solutions that best hedge these flows under both our methodology and that of KRD-matching.
  • We analyze these solutions relative to the actual performance of the liabilities, both in terms of their tracking error and their ability to sustain funded status.
  • Our solutions featured very favorable information ratios relative to the KRD Match solution.
  • Analysis here shows that the apparent on-paper precision of KRD-matching is less compelling both conceptually and in practice than might be widely believed.
  • The empirical runs in this paper suggest that our process can perform well in “real world laboratory” conditions.

Section I. Introduction

Highly customized liability-driven investing (LDI) solutions are most appropriate when a defined-benefit (DB) pension plan has achieved its desired funded status level and wants then to reduce tracking error risk against liabilities as much as possible. While a “standard” LDI allocation of long credit or long G/C can remove most mismatch in overall asset duration versus liability duration, customized solutions seek to further reduce tracking error by hedging specific maturity ranges of the liabilities.

The question becomes how best to achieve such a minimum-risk allocation. Many plans are customized by attempting to match all key-rate durations (KRDs) of the liabilities—that is to hedge all the maturity ranges affecting liability valuation. At Western Asset, our approach seeks to minimize tracking error by using empirical data to tell us which maturity ranges to hedge most strenuously and which combination of credit and Treasuries to use to effect this hedge.

KRD-matching looks great on paper. However, there are real-world drawbacks to its efficacy. KRD-matching alone provides no guidance as to how to mix credit and Treasuries in a solution. The implicit belief is that a year of duration—or KRD—in either Treasuries or A or BBB rated credit will behave the same as a year of duration—or KRD—in AA DB liabilities. In actual practice, credit spreads are constantly varying, so that merely matching duration or KRDs will still result in substantial basis risk between assets and liabilities. Similarly, KRD-matching provides no guidance as to how to provide sufficient asset returns to match those of the liabilities.

Again, our approach uses historical data to determine how to address all these issues. It’s an empirical question as to which approach works best. This paper runs “real-world” tests to answer this question.

Utilizing an actual set of client liability cash flows and their evolution over time, we identify the customized solutions that best hedge these flows under both our methodology and that of KRD-matching. We analyze these solutions relative to the actual performance of the liabilities, both in terms of their tracking error and their ability to sustain funded status. We construct these solutions utilizing a variety of candidate assets.

The Appendix to this paper presents the analytics underlying KRDs in detail. Section 2 summarizes the key points of that analysis. Section 3 discusses our own approach. Section 4 reports the results of our performance simulations. Section 5 provides our conclusions.

Section 2. KRD-Matching?

KRDs are calculated by determining how shocks to segments of the Treasury par curve result in shocks to the corresponding spot curve and how these changes in spot yields affect prices of Treasury and credit bonds. Because the price of a couponed bond is said to depend on all spot yields with equal or lesser maturity, some bonds will exhibit KRD against all key rates with equal or lesser maturity. As a result, low-maturity KRDs can be matched via long bonds. So, liability KRDs can usually be fully matched with a portfolio largely confined to long-duration assets.

This is how KRD-matched solutions typically achieve their hedges. A number of questions arise, in addition to the problems with KRD-matching cited in the previous section. Can 20- and 30-year bonds really do a good job of hedging real-world fluctuations in yields at, say, the 2-, 5- or 10-year maturity points? Alternatively, how plausible is it that the prices of 20- and 30-year bonds are sensitive to movements in spot yields in the 2-, 5- and 10-year maturity ranges?

This is certainly plausible for Treasuries, where an abundance of both couponed and stripped issues of identical credit quality are actively traded in the market, so that arbitrage can keep spot yields and yields-to-maturity in harmony with each other. However, corporate bonds trade with wildly different quality, and typically, no individual issuer has more than a few bonds in trade. Furthermore, KRD practice is that only a single option-adjusted spread (OAS) is added to the spot yields for the various cash flows to determine its price. It is much more of a stretch to assert that market arbitrage will keep all corporate bonds’ yields in line with Treasury spot yields. Yet, the efficacy of KRD-matching depends on that premise.

Furthermore, the elegant mathematics of KRD calculations prescribes that bonds of identical quality and maturity can have radically different KRD profiles depending on whether they are priced above, at, or below par. Because changes in a particular key par rate change no other par yields but change all spot yields of equal or greater maturity, Treasury bonds priced at par exhibit KRD only to key rates of similar maturity. Treasuries priced at premium exhibit positive KRDs against all key rates of equal or lesser maturity, while those priced at discount exhibit negative KRDs against such key rates.

Exhibit 1 illustrates these results by showing actual KRD patterns for A-or-better credit indices and for Treasuries as of October 31, 2015 (the date on which our runs will begin). As seen there, the long-duration indices exhibit positive KRD against all key rates of equal or lesser maturity, while STRIPS, being a heavily discounted bond, exhibit negative KRDs against lower-maturity key rates.

Exhibit 1: Key-Rate Durations of Liabilities and Fixed-Income Assets as of 10/31/15
Key-Rate Durations of Liabilities and Fixed-Income Assets as of 10/31/15
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

This begs the question of whether bonds with identical maturity and quality really exhibit such varied price behavior in the real world. Our tests will provide some perspective as to how well these solutions perform relative to our own.

Section 3. Our Approach

Our approach utilizes empirical data to identify optimized custom solutions. Given the client’s liability cash flows, we simulate the historical behavior of liability returns by determining how the valuation of those flows would have changed in response to actual, historical movements in the discount method used to evaluate those liabilities.

Exhibit 2: Efficient Frontier, Funded-Status Optimization (LDI)
Efficient Frontier, Funded-Status Optimization (LDI)
Source: Western Asset. Select the image to expand the view.

This sample of “historical” liability returns can then be compared to historical behavior of the relevant candidate asset indices over the same period to identify those combinations of assets that best reduce tracking error or best produce adequate returns. In effect, this process constructs an efficient frontier of asset/liability solutions, as depicted by Exhibit 2.

The client then chooses the point on that frontier that satisfies their return targets. This process can be used for any DB plan at any funded status and with any mix of “hedging” and “return-seeking” assets. For fully funded plans seeking to reduce tracking-error risk to an absolute minimum via a customized solution, a candidate set of suitable fixed-income assets is specified, and from that set a solution is chosen on the efficient frontier that sustains funded status (i.e., for which expected assets' returns equal or exceed expected liability returns).

This process automatically determines the optimal mix of credit and Treasuries. Expected returns are explicitly targeted to match a client’s objectives. Finally, this process, by construction, hedges the KRDs most responsible for historical variation in liability returns.

An obvious objection to our approach is that past performance is no guarantee of future results. There is no guarantee that a hedge that worked well in simulations over historical data will also work well in the future in real time. That same objection could be made to any efficient-frontier analysis or to any usage of correlations. Yet, efficient frontiers and correlations are routinely used in finance.

Our approach relies on the premise that the future will reasonably resemble the past. The KRD-matching approach relies on a premise that has never held in actual experience: that credit spreads will be invariant across different qualities, so that A and BBB rated bonds (and Treasuries) of a given duration (and KRD) will move in virtual lockstep with AA rated liabilities with the same duration (and KRD). Again, the efficacy of either approach is an empirical question. So, let’s look at the empirical data.

Section 4. Comparing Solutions

Our primary “horse race” between the two approaches were conducted in terms of the actual cash flows of a particular client as they have evolved over time. On a number of occasions, the plan actuary completely re-estimated liability cash flows for various reasons. At each such rebalance point, we calculated new customized solutions both for our approach and for the KRD-matching approach against these flows.

Exhibit 3: KRD Profiles For 10/15 Solutions, Discrete-Maturity Credit
KRD Profiles For 10/15 Solutions, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

Our solutions were constructed optimizing over a back-sample of the previous five years of simulated historical liability returns against actual returns on candidate assets. The KRD-matched solutions were calculated using KRDs holding for both liabilities and candidate assets as of the date of the rebalance. Resulting custom solutions were then tracked using actual asset returns against actual liability returns for each month until the liabilities were again re-estimated, at which point this process was repeated.

All told, there were 13 liability re-estimations and resulting solution rebalancings, with the most recent tracking through August 2021. Each such rebalancing utilized only information available as of the date of the re-estimation: the cash flows in question, the preceding history of assets and liability discount curves, and the KRDs in place as of the re-estimation date. So, this is as close to “real-time” tracking as we could get ex post with available data.

Our primary Optimize Return solution attempts to minimize tracking error subject to achieving an average return equal to or greater than that of the liabilities. In addition, we also constructed two other solutions under our approach: one that minimized tracking error without regard to return (Minimize TE) and one that minimized tracking error subject to the constraint that portfolio yield at least equal the liability yield at the point of rebalancing (Yield Optimized).1 These provide some perspective on our primary solution.

Besides calculating a simple match of KRDs (KRD Match), we also calculated a solution that best matched KRDs subject to requiring portfolio yield to at least equal liability (KRDs with Yield Match).

Given the actual formulation of KRDs (detailed in the Appendix), it will often be the case that exact matching of KRDs cannot be achieved without shorting some issues. So, for our runs, which did not allow shorting, “KRD-matching” solutions were taken to be those that minimized the sum of squared deviations of asset versus liability KRDs across the various key rates.

In Exhibit 4, “RMSE for KRD Mismatch” measures the square root of this sum of squared deviations for each solution.2 The lower the RMSE, the tighter the match of KRDs. Since KRD's deviations are summed without weightings, this minimization treats all KRDs as equally important. (In contrast, our approach implicitly focuses on which KRDs have been most responsible for volatility in liability returns.)

Section 4.1. Simulation Runs With Discrete-Maturity Credit and Treasuries

The first run we performed with these liabilities constructed solutions from candidate assets consisting of a suite of discrete-maturity A-or-better credit indices and 25+ year STRIPS and 7-10 year governments. Credit indices were constructed with maturity ranges of 0-3.5 years, 3.5-7.5 years, 7.5-15 years, 15-25 years and 25-30 years to roughly match the standard key-rate maturities. The idea was to let these credit indices do the heavy lifting in hedging important maturity ranges, with Treasuries supplementing.3

Exhibit 3 shows the KRD sets for the liabilities as of October 31, 2015 and for the various solutions. While it is hard to detect on the chart, even the KRD Match solution produced slight mismatches of the KRDs. Its overall duration was 12.05 years versus 11.79 years for the liabilities.

Once a minimum yield constraint was added to the KRD-matching process, the resulting solution featured higher exposure in the 10-, 20- and 30-year key rates maturities in order to boost yield. For our solutions, minimizing tracking error with or without constraints, allocations to 10- and 30-year key rates were underweighted in order to focus more on the 20-year area, as that maturity range was found to have contributed more of the volatility in liability returns occurring over the sample period just prior to October 2015.

Exhibit 4 shows these solutions for the October 2015 rebalancing, as well as summary statistics from the estimation process. Portfolio Yield Difference shows the amount by which the solution’s yield exceeds or falls short of liability yield as of October 31, 2015. Average return and tracking error are those over the back sample.

Exhibit 4: Customized Indices as of Oct 2015, Discrete-Maturity Credit
Customized Indices as of Oct 2015, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.
The returns contain simulated past performance. Simulated past performance is based on the actual past performance of one or more financial instruments or financial indices which are the same as, or substantially the same as, or underlie, the financial instrument concerned. Past performance is not a reliable indicator of future performance. 4
Exhibit 5: Optimize Return Solutions, Discrete-Maturity Credit
Optimize Return Solutions, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.
Exhibit 6: KRD-Match Solutions, Discrete-Maturity Credit
KRD-Match Solutions, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

The KRD-matching solutions typically feature heavy allocation to long-duration A-or-better corporates and little or nothing to Treasuries. For the solutions utilizing our approach, allocations were focused on the 15-25 year A-or-better bucket and less so on the other corporate buckets, and there was somewhat more allocation to Treasuries. Exhibits 5 and 6 illustrate these points by showing the evolutions of our Optimize Return solution and the KRD Match solution across rebalance dates.

Exhibit 7 shows summary statistics for the performance of each solution against the liabilities over the whole simulation period, 11/15-8/21. No one solution exhibited both higher returns and lower tracking error than the others. The KRD Match solution produced relatively low tracking error, but it also underperformed liabilities by 3.5 bps per month on average (for a 245-bp underperformance over the sample). Constraining the portfolio yield to match or exceed liability yields, while also matching KRDs, increased returns by 6.8 bps per month, at a “cost” of an extra 8.7 bps per month of tracking error.

Our Minimize Tracking Error solution indeed produced the lowest tracking error. While it also underperformed the liabilities on average, its performance dominated that of the KRD Match. For the other solutions, both average miss (excess return) and tracking error were higher than for the KRD Match.

Exhibit 7: Summary Statistics for Customized Solutions, Discrete-Maturity Credit
Summary Statistics for Customized Solutions, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

Information ratios (IRs) measure the increase in return divided by increase in risk, in this case relative to the KRD Match solution, as shown in Exhibit 7. The Optimize Return solution, our standard product, produced far and away the highest IR, at 2.1, with a tracking error only slightly above that of the KRD Match and an average return substantially higher.

Finally, Exhibit 8 shows actuarial year aggregate misses relative to liabilities for the five solutions. “Interim months” is the period 11/19-3/20, when the plan transitioned from an actuarial year ending in October to one ending in March. As seen there, the various solutions tended to over- or under-perform the liabilities together in each actuarial year, reflecting common sensitivities to yield and spread movements.

Exhibit 8: Annual Performance of Custom Solutions, Discrete-Maturity Credit
Annual Performance of Custom Solutions, Discrete-Maturity Credit
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

Section 4.2. Simulation Runs with Broad Credit and a Discrete-Maturity STRIPS

When we presented an earlier run of the Section 4.1 results to an investment consultant team that works on KRD-matching solutions, they commented that rather than using a suite of discrete-maturity credit instruments, they instead used the broad long credit index with a suite of discrete-maturity STRIPS. In order not to exceed clients’ specified limits on BBB exposure, they used A-or-better credit, and in order to better match the yields and returns of the liabilities, they required a minimum 80% exposure to credit. Finally, they said they always kept the hedge ratio (asset duration divided by liability duration) at 100%.

To incorporate these specifications, we constructed the following candidate asset set: (1) Long A-or-better Credit, (2) Intermediate A-or-better Credit, (3)-(7) STRIPS with maturity ranges 1-2, 4-5, 9-10, 19-20 and 29-30 years in maturity, these maturity ranges clearly corresponding to the key rate maturities. With this set of candidate assets, the same five solutions were constructed and simulated as before, with credit comprising at least 80% of assets for the KRD-matching solutions. Finally, for the KRD Match solution only, we required that the hedge ratio round to 100% (be greater than 99.5% and less than 100.5%).

For this particular client’s cash flows, the length (duration) of the flows declined over time, with plan mergers and actual experience. For the solutions under both our and the KRD approaches, credit exposure gradually shifted to some extent from long credit to intermediate credit, in reflection of this shortening of liabilities. The KRD Match solutions generally held credit exposure to the required 80% minimum. Our solutions typically allocated about 95% of assets to credit.

Exhibit 9 summarizes the results. Here, the KRD Match solution produced the lowest tracking error, but underperformed liabilities by almost as much as in the previous run. The other four solutions all produced both higher average returns and higher tracking errors. Here again, our standard, Optimize Return solution produced the highest information ratio vis-à-vis the KRD Match, with an extra 3.7 bps per month of return versus an extra 3.2 bps per month of tracking error, for an information ratio of 1.1.

Exhibit 9: Summary Stats for Customized Solutions, Credit & Discrete-Maturity STRIPS
Summary Stats for Customized Solutions, Credit & Discrete-Maturity STRIPS
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

Section 4.3. Simulation Runs with Credit and “Futures”

The runs so far featured all physical (cash) instruments for candidate assets. A third run incorporated the use of derivatives in customized solutions. A problem arises as to how to choose among contract expiry dates. Also, KRDs for futures contracts are dependent upon which cash bonds are cheapest to deliver, something a manager can’t know exactly ahead of time.

To provide objective, “data blind” instruments, we constructed quasi-futures contracts, overlays that were long Bloomberg benchmark issues and short 1-month Treasury Bills (T-Bills), with benchmark indices chosen to match the maturities of existing futures contracts: 2, 5, 10 and 30 years. We required holding at least as many T-Bills as was necessary to fulfill initial margin requirements for these contracts.

The same five solutions were constructed as before. For these runs, the KRD Match solutions allocated only the minimum 80% to credit, with 20% of assets invested in T-Bills. In contrast, the solutions for our approach typically held only as many T-Bills as was necessary to meet margin requirements.

Obviously, 80% allocation to credit is more than sufficient to match the KRDs of the liabilities when derivatives are allowed. However, it is not generally enough to match liability returns and yields, thus the greater exposure to credit for our solutions.

As seen in Exhibit 10, both KRD-matching solutions substantially underperformed the liabilities on average. In fact, of the five solutions, only our Optimize Return solution was able to match average return on liabilities. So, it would not appear that the ability to invest in derivatives was a boon to asset returns over this sample. Similarly, the tracking errors displayed in Exhibit 10 are not generally lower than those in Exhibit 9.

As with the two preceding runs, our solutions exhibit very favorable information ratios against the KRD Match solution.

Exhibit 10: Summary Stats, Customized Solutions, Credit/Futures
Summary Stats, Customized Solutions, Credit/Futures
Source: Bloomberg, Client’s Actuary, Western Asset. Select the image to expand the view.

Section 5. Conclusions

Besides the comparison runs described in Section 4, using cash flows that evolved over time, we also performed other runs using static liability cash flows, with liability duration levels both equal to and higher than those used in Section 4. The results of those runs were essentially identical to those in Section 4.

Obviously, the simulations described here are not exhaustive. They are specific to the liability sets used and the market conditions encompassed by our 2015-2021 sample period. Still, it is clear that our approach more than held its own against the KRD-matching solutions.

While the KRD-matched runs occasionally produced slightly lower tracking error, they were invariably unable to match the average returns of the liabilities or of our solutions. Our solutions featured very favorable information ratios relative to the KRD Match solution.

Granted, the focus of customized solutions is to minimize tracking error against the liabilities, but among real people, a “constant” underperformance is regarded as “tracking error” even if the statistical volatility around that negative mean is low or zero. To put it differently, a steady decline in funded status is not acceptable performance, even if that steady decline registers as (slightly) lower tracking error than a solution that sustains funded status.

Purveyors of KRD-matched solutions argue that any bleed in funded status within a customized fixed-income allocation can be offset by investments in equities or other “return-seeking” assets. However, that ignores the whole purpose of constructing customized LDI solutions. Again, the intent is to minimize tracking error as much as possible. Yet, the introduction of equities and other risk assets will inevitably increase overall tracking error by more than customizing hedge assets reduces it.

Elsewhere,5 we have argued that customized solutions for under-funded plans or plans with substantial risk-asset allocations are “sweating the small stuff,” taking great pains to slightly reduce risk with the right (LDI) hand while the benefits are squandered by the left (risk asset) hand. This is not an effective comprehensive approach.

So, it matters whether the returns from even a customized LDI portfolio are sufficient to match liability returns. In the runs here, our solutions performed well on this score, with little or no increase in tracking error relative to KRD-matched solutions and uniformly higher returns.

Meanwhile, in comparing the results across the three runs, it is interesting that those solutions allocating to broad credit and futures did not produce any general reduction in tracking error nor any increase in average returns relative to those allocating to broad credit indices and STRIPS. However, the solutions allocating to broad credit and STRIPS did produce lower tracking errors than those allocating to discrete-maturity credit and Treasuries.

The apparent precision of KRD-matching might be too attractive for some plans to pass up. However, the analysis here has shown that the apparent on-paper precision of KRD-matching is less compelling both conceptually and in practice. And the empirical runs here at least suggest that our process can perform well in “real-world laboratory” conditions.

View the appendix

  1. The yield on liabilities is identified as the discount rate—or internal rate of return—for the liabilities at any point in time. This is the yield level that produces the same liability valuation as is produced by the spot curve used to evaluate the liabilities.
  2. RMSE stands for Root Mean Squared Error.
  3. The Minimize Tracking Error and KRD Match solutions impose no constraints on asset returns or yields, and the resulting solutions for these runs were uniformly well-diversified combinations of the candidate assets. For the other runs, at times, historical liability returns and/or liability yields were so high relative to those of the assets that the only solution satisfying the strict return or yield constraints were undiversified mixes primarily consisting of the asset with highest historical return or highest current yield.
    In actual experience, when this has occurred, our practice has been to slightly ease return constraints in order to achieve a more diversified portfolio. We followed this same practice in the runs here. For the Optimize Return solution, the 4/17 and 10/18 runs required some easing of constraints, and the solution was chosen that minimized tracking error subject to the requirement that average asset returns were no more than 2.0833 bps per month (25 bps per year) below that of liabilities. For the Optimize Yield and KRD with yield runs, the 10/19 and 3/21 rebalancings required an easing of constraints, and we chose solutions with portfolio yield no more than 25 bps below that of the liabilities. In each case, the hope was that experienced returns below those of the liabilities would be minor until the next rebalancing, at which point it was expected that the constraints would be more easily met. This was indeed the case for the return-optimized runs.
    The “eased” constraints then were in place over eight months of the 70-month sample for the return-optimized solution and 10 months of 70 for the yield-constrained solutions. As with the other rebalancings, no information was used in constructing these solutions other than that available as of the rebalance date and that detailed in this note.
  4. Simulated portfolios have inherent limitations. Material economic and market factors may affect investment decisions differently when the managers are investing actual client assets. Simulated portfolios do not reflect the impact of actual portfolio trading, which may affect the price and availability of securities. The hypothetical performance returns presented for the simulated portfolio do not include the deduction of management fees, transaction costs, or other account expenses and assume the reinvestment of income and capital gains.
  5. See Effective LDI: Don’t Sweat the Small Stuff, April 2014, available on our website.