• Standard hedge ratio calculations compare the dollar duration of DB plan assets to that of the plan’s liabilities, with no consideration of the different credit quality or of the different yields that drive asset and liability valuations.
  • Corporate bonds exhibit much less “empirical duration” and much more “empirical spread duration” than standard duration measures suggest.
  • Equities and other risk assets typically exhibit substantial, negative empirical rate duration and substantial, positive empirical spread duration, even though they are usually assumed to have zero duration.
  • Because of these discrepancies, managing a typical DB plan according to standard hedge ratio metrics will leave it dramatically less hedged against movements in yields and dramatically more sensitive to movements in credit spreads than its managers realize.
  • We formulate and report effective hedge ratio and effective spread hedge ratio measures based on the real-world performance of assets and DB liabilities.
  • Our effective hedge ratio metrics can be used to produce DB allocations that more effectively hedge DB liabilities while also targeting adequate returns.

Many defined-benefit (DB) pension funds zealously target and track their hedge ratio: the ratio of the dollar duration of their plan assets to that of their liabilities. They believe this measure accurately reflects the extent to which the interest-rate sensitivity of their liabilities is hedged by the interest-rate sensitivity of their assets. In actuality, the accuracy and utility of standard hedge ratio measures depend on assumptions about yields and credit spreads that are far removed from reality.

We believe standard hedge ratio measures seriously misstate the true hedging capability of most DB asset allocations. At Western Asset, we formulate and report an effective hedge ratio measure that better reflects the relative sensitivity of assets and liabilities to real-world market conditions. We also formulate and report an effective spread hedge ratio measure that reflects the relative sensitivity of DB assets and liabilities to real-world movements in credit spreads.

For standard DB asset allocations, effective hedge ratios will typically be sharply lower than standard—“on-paper”—hedge ratios. In other words, DB plans are typically much less hedged against yield movements than their managers realize. Similarly, effective spread hedge ratios are typically dramatically higher than standard spread hedge ratio measures suggest, indicating much higher exposure to spread movements is typically realized.

These differences between standard/on-paper and effective hedge ratio measures arise largely because the yields of different financial market instruments do not move in lockstep together. They are not perfectly correlated. When yields at different quality points diverge, bonds with equal duration will exhibit substantially different changes in valuation, so that one will not be a good hedge of the other. On-paper duration and hedge ratio calculations do not allow for this.

Similarly, while equities and other “risk assets” are assumed by standard formulations to exhibit zero duration and zero spread duration, in practice, risk assets exhibit substantially negative correlation with Treasury bond prices and substantially positive correlation with corporate bond prices. This also affects the hedging capability of DB allocations in ways that standard hedge ratio calculations fail to reflect.

On-Paper and Effective Hedge Ratios For a Sample DB Allocation

Exhibit 1 shows analytics for a typical DB plan. The plan’s “hedge portfolio” is diversified, but with only a small allocation to Treasuries. Its “return-seeking” portfolio is also diversified, with exposure to both domestic and global equities, as well as various “alternative” assets. The plan has made some efforts to reduce risk, as reflected by a 55% allocation to fixed-income and a reported 58% hedge ratio.

Exhibit 1: On-Paper and Effective Hedge Ratios for a Typical DB Plan, 100% Funded Status
Source: Barclays, Cambridge Associates, Hedge Fund Research Institute, Ibbotson, MSCI, ODCE, Western Asset. As of 31 Dec 18. Select the image to expand the view.

However, in actual practice, we find that the effective hedge ratio for this plan is only 12%, so that the plan will experience disruptive variations in funded status when and as Treasury yields vary meaningfully. Similarly, we find this plan to exhibit a 210% effective spread hedge ratio, compared to a stated spread hedge ratio of only 49%. So, the plan’s funded status will be dramatically more sensitive to credit spread movements than conventional metrics indicate.

These sensitivities show up clearly in a backtest of the asset portfolio against plan liabilities. As seen in Exhibit 2, asset returns diverge sharply from liability returns in a number of years, most notably in 1990-92, 1994-96, 1999-2004, 2006-09, 2013 and 2017, this, again, for a plan that is reportedly 58% hedged. Overall, this plan exhibits high, 8.8% funded status volatility (tracking error between assets and liabilities). Even though the compound average return on assets matches that of the liabilities over this period, the need to make benefit payments during “drawdown periods,” such as 1991-92, 2000-03, or 2008 would have resulted in the plan steadily losing funded status over time.¹

These are the maladies associated with high tracking error. They result from a low hedge of liabilities’ yield sensitivity and from excessive sensitivity to credit spreads. Yet, standard hedge ratio metrics fail to accurately reveal the extent of these mismatches.

Exhibit 2: Backtest of Typical Allocation vs. Liabilities Over 1989-2018
Source: Barclays, Cambridge Associates, Hedge Fund Research Institute, Ibbotson, MSCI, ODCE, Western Asset. As of 31 Dec 18. Select the image to expand the view.

Hedge Ratios in Principle and in Practice: Liability Driven Investing (LDI) Meets Modern Portfolio Theory (MPT)

Standard duration calculations measure the sensitivity of the prices of various fixed-income instruments to changes in the yields of those instruments. However, the prices of different instruments depend on different yield measures. So, taking the relative durations of two bonds as an indicator of their relative sensitivity to yield changes will work as planned only if both their yields fluctuate by the same amount. This almost never happens in the real world.

It is well known that for Treasury instruments of different maturities, their durations will accurately describe their relative price sensitivity only opposite parallel shifts in the yield curve. For exactly the same reasons, bonds with the same duration but different credit quality will exhibit equal price changes only if credit spreads are invariant. Their relative durations will be an inaccurate measure of their “rate-sensitivity” whenever credit spreads change.

And spreads do change. Typically, when Treasury yields fluctuate, credit spreads move in the opposite direction. They are negatively correlated with Treasury yields. Furthermore, the lower the quality of the instrument, the less its yield and price tend to move in sync with those of Treasury instruments.

All this detracts from the utility of standard hedge ratio measures. Again, these measures simply divide the total (dollar) duration of assets by that of liabilities, with no regard for the respective credit qualities of assets and liabilities. DB pension liabilities are evaluated, for accounting purposes, via AA yields, so they effectively display AA credit quality. However, given the paucity of actively traded AA bonds, DB asset allocations are typically dominated by exposure to A, BBB or even speculative-grade bonds.

The on-paper durations of these bonds will grossly overstate their sensitivity to real-world changes in the AA yields that drive liability valuations. To put it differently, the less-than-perfect correlation between asset yields and liability yields results in basis risk that standard hedge ratio calculations do not reflect.

Furthermore, blithely assigning zero duration and zero spread duration to equities and other risk assets does not prevent them from exhibiting significant correlation with changes in Treasury yields and credit spreads, and this is another source of basis risk for DB allocations. Similarly, holdings of foreign-currency bonds in US DB allocations are typically assigned durations corresponding to their sensitivity to foreign-currency yields. Not only do spread movements come into play here, but exchange-rate movements do as well, both serving to reduce the meaningfulness of standard hedge ratio calculations.

If the references to correlation here remind the reader of MPT analysis, they should! We use empirical correlations to assess the effective hedging power of DB allocations, and this is nothing more than an application of MPT to LDI and liability hedging.

Over the past 60 years, MPT has revolutionized investment analysis and portfolio construction. It is hard to see a serious analysis of an equity portfolio that does not mention the “beta” of the portfolio or its components, or the correlation of those elements with each other and with various market factors. Yet, these same, now-basic MPT tools are often completely absent from fixed-income or DB pension analysis. Fixed-income analysis has always been highly quantitative, employing tools such as yield, duration, convexity, etc. but these tools are often applied inaptly, where the assumptions empowering them (i.e., parallel yield curve shifts, constant credit spreads) are too far removed from reality to make valid inferences.

On-Paper and Effective Durations for DB Candidate Assets

Our construction of LDI allocations is based on usage of real-world data to describe the sensitivity of liabilities and assets to various market factors. Our LDI solutions are those asset allocations that minimize funded status volatility over a representative sample of real-world data, subject to return objectives acceptable to our clients.

We produce “empirical” or “effective” duration and hedge ratio calculations that reflect real-world market conditions. Effective (rate) duration should measure the sensitivity of an instrument to real-world movements in basic yields, to Treasury yields. Effective spread duration should measure the sensitivity to real-world movements in credit spreads. Of course, once durations and spread durations are to be defined in terms of real-world market movements, some standards of rate and spread movements must be stated (just as the betas for assets must be defined relative to some standard of market movements).

We consider the Bloomberg Barclays Long Government Index to be the standard for rate movements. We take its duration to be as stated, and we measure the effective rate duration of other instruments as their “betas” against the Long Government Index times the duration of long governments.

Similarly, we consider the Bloomberg Barclays Long Credit Index the standard for credit spread movements, and we take its effective spread duration to be as stated. We measure the effective spread duration of other instruments as their betas against long credit times the duration of long credit.²

For portfolios, the effective rate and spread durations are merely weighted averages of the effective durations and spread durations of the components, as with standard measures. Similarly, effective rate and spread hedge ratios are the ratios of the effective durations of assets and liabilities (i.e., dollar-durations, with effective durations weighted for the respective sizes of assets and liabilities).

Exhibit 3: On-Paper and Effective Durations
Source: Barclays, Cambridge Associates, Hedge Fund Research Institute, Ibbotson, MSCI, ODCE, Western Asset. As of 31 Dec 18. Select the image to expand the view.

Exhibit 3 compares our effective estimates to standard metrics for a range of asset instruments. Again, the effective duration of long governments is taken to be its on-paper (reported) value, while the effective spread duration of long credit is taken to be its on-paper value. For credit instruments, effective duration is well below on-paper values, and the lower the quality, the bigger the disparity between on-paper and effective duration. This reflects the negative correlation between Treasury yields and spreads, as discussed earlier.

It is interesting that Treasury STRIPS uniformly exhibit more effective duration than their on-paper values suggest. This might reflect the greater convexity of STRIPS and/or it might reflect a larger movement in STRIPS yields than in coupon yields when long Treasury yields move.³

As mentioned previously, risk assets generally exhibit negative effective duration, real estate being the only exception. Non-dollar equities show more negative effective duration than do US equities.

Not surprisingly, the lower the credit quality, the more effective spread duration the instrument displays. (Compare high-yield to intermediate credit or long credit to the long-maturity A or better index.) Risk assets generally exhibit substantially positive effective spread duration, again with real estate being the only exception. STRIPS exhibit some nonzero effective spread duration. This may be a spurious correlation, but it shows up in our sample, so we report it here.4

Effective Hedge Ratios For Well-Constructed LDI Solutions

We’ve presented and analyzed the differences between reported and effective durations and spread durations. The benefits from this analysis accrue by applying the insights learned to the process of constructing more risk-efficient LDI allocations. The gulf between reported and effective hedge ratios can be narrowed by better balancing fixed-income assets between governments and credit. Excessive sensitivity to credit spread movements can be addressed by understanding the sensitivity of risk assets to credit spreads and, here too, blending in more exposure to governments to balance the effective spread duration provided by most risk assets.

Exhibit 4: On-Paper and Effective Hedge Ratios for an Optimized Allocation, 100% Funded Status
Source: Barclays, Cambridge Associates, Hedge Fund Research Institute, Ibbotson, MSCI, ODCE, Western Asset. As of 31 Dec 18. Select the image to expand the view.

Of course, more exposure to Treasuries means downward pressure on asset returns. However, our process works to ensure that allocations to credit and risk assets are made on a risk-effective basis. Also, besides attempting to minimize risk, our process explicitly addresses asset returns to make sure that resulting solutions fulfill clients’ return targets. A sample of this process can be seen by taking the same assets and liabilities as were discussed previously for the “typical DB plan” (Exhibit 1), but allocating to them more efficiently.

With the same assets and liabilities in play as detailed in Exhibit 1, our LDI Optimizer model finds that the allocation shown in Exhibit 4 provides minimum tracking error against liabilities while also matching the expected return on liabilities. Not surprisingly, the optimized allocation features a much heavier exposure to long governments than in Exhibit 1 (43%, up from 4%). Somewhat more assets are allocated to high-yield as well (11%, up from 5%).

Again, this allocation provided sufficient average returns to match those of the liabilities. Average returns over the sample period were only 24 basis points (bps) per year lower than for the sample allocation in Exhibit 1. Yet, the gap between reported and effective hedge ratios has been narrowed dramatically. While the effective spread hedge ratio is still above 100%, it is dramatically less so than what the sample portfolio exhibited (116% down from 210%).

As seen in Exhibit 5, returns on the “optimized” allocation track much more closely to liabilities than was seen in Exhibit 2. Of course, there is still tracking error, but much less than before: 2.2% here versus 8.8% for Exhibit 2. Also, there are no runs of continued under- or over-performance of assets against liabilities, such as were seen for the sample portfolio in Exhibit 2.

Exhibit 5: Backtest of Optimized Allocation vs. Liabilities Over 1989-2018
Source: Barclays, Cambridge Associates, Hedge Fund Research Institute, Ibbotson, MSCI, ODCE, Western Asset. As of 31 Dec 18. Select the image to expand the view.

Further improvements could be achieved by allocating to customized LDI instruments, such as the STRIPS and fixed-maturity instruments shown at the bottom of Exhibit 3, as well as to derivatives. With such instruments, the allocation could be fine-tuned to the point that on-paper and effective hedge ratios were nearly equal, and tracking error could be reduced somewhat further. However, the results in Exhibit 4 hint at how much performance can be improved merely by allocating to existing assets more efficiently, resulting in effective rate hedge ratios much closer to 100% and effective spread hedge ratios much lower than typical allocations display.


Standard hedge ratio metrics are accurate only if the extreme assumptions of parallel yield curve shifts and invariant credit spreads hold. Real-world behavior differs so markedly from these assumptions that standard hedge ratio metrics are of dubious practical value.

Our LDI solution process arises from an analysis of actual, empirical financial market data. It essentially applies modern portfolio theory to the task of pension optimization. The details of this process provide effective hedge ratio and effective spread hedge ratio metrics that we believe are dramatically more useful to our clients than standard (on-paper) measures.

Our choices of long government and long credit to gauge prototypical rate and spread movements are arbitrary, but these are the market indices most relevant to DB plans. Furthermore, any other standards, when properly defined, would provide relative measures of effective hedge ratio essentially the same as ours.

As always, past performance is no guarantee of future results. Our approach will be effective only if future experience is roughly comparable to that of the past. Then again, zealously following on-paper hedge ratio metrics is accurate only if the future is unlike anything ever seen before: a world of invariant credit spreads and/or parallel yield curve shifts. Our process depends no more on past performance than does any other application of modern portfolio theory.

Our custom LDI solutions have indeed performed well against client liabilities in real time. Hopefully, the effective hedge ratio and effective spread-hedge ratio concepts introduced here will provide useful insights for our clients on the efficacy both of their current allocations and of potential LDI solutions. Generally, if your DB plan contains substantial allocations to credit and/or risk assets, you are likely much less hedged against rate movements and much more sensitive to spread movements than you think.

  1. The problems of sustaining funded status when benefit payments must be made were explored in Why All Defined-Benefit Plans are Short-Term Investors, November 2014, available on our website.
  2. More precisely, the standard for interest and spread movements are the risk factors we define for the long government and long credit markets. Those risk factors were introduced and described in A Risk Factor Based LDI Analysis: A 100% Hedge Ratio Is Not Enough, May 2015, available on our website. These risk factors are derived in a way that makes them uncorrelated with each other. The risk factor for the long government market measures movements in long government returns that are not correlated with movements in the T-bill market. The risk factor for long credit measures movements in long credit returns that are not correlated with movements in the T-bill, long government or intermediate government markets. Our process identifies seven risk factors for LDI analysis: short, intermediate and long government market factors, short and long credit market factors and large and small cap equity market factors. These seven factors are akin to the five factors of the well-known Fama-French model, but are derived in quite a different fashion.
  3. For a more detailed discussion of convexity in LDI, see Convexity in LDI Liabilities—How to Earn More and Be Better Hedged, November 2017, available on our website.
  4. These effective duration and spread duration measures were calculated using annual holding period data over 1989-2018, the same sample as displayed in Exhibit 2. We also note here that while the effective rate and spread durations of high-yield are about equal to those of large cap equities, the latter is also sensitive to equity-market specific risk factors, so that equities indeed exhibit more volatility and risk than high-yield.