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Effective LDI: When Do Bond Swaps Help?

Western Asset

Executive Summary

  • Overlaying an interest rate swap onto an equities-heavy pension allocation does little to reduce surplus volatility, and under some circumstances, it could raise surplus volatility.
  • Extending the duration of fixed-income assets is simpler than a swap, and it is more likely to provide some benefit.
  • Plans must be aware of the trade-off between long-run returns and short-run stability when formulating their asset allocation.

I. Introduction

Your company’s defined-benefit pension plan features an asset allocation heavily concentrated in equities. It has been proposed that you overlay the portfolio with a long-duration interest rate swap, so as to eliminate interest rate risks from your plan. Is this a good idea?

Probably not. While such tailored swaps can reduce interest rate sensitivity, they are costly and illiquid. At best, they will only marginally reduce plan surplus volatility for an equity-heavy allocation, and at worst, they could raise surplus volatility.

Overlaying a swap does work to raise the duration of plan assets. The optimal swap exposure would match the total (dollar) duration of plan assets to that of plan liabilities. However, surplus volatility also arises from non-interest factors, and with asset allocations devoted 60% or more to equities, the non-interest risks of assets dominate the interest risks of liabilities.

A simple LDI alternative would be merely extending the duration of fixed-income assets. Such a tactic avoids the complexities of the swap and is virtually sure to reduce surplus volatility. While this is a small step, it is a simple, effective tactic.

Interest-rate swaps have limited effectiveness as LDI tools because of the dominance of equities in pension allocations. Equities are inherently risky in the short-run. Pension plans invest heavily in equities because of the long-term benefits, and that is fine. However, a plan cannot expect to achieve both long-term return enhancements via equities and short-term surplus stability. There are no magic bullets that will hit both goals.

Bond Swaps and Surplus Volatility

A pension plan’s funded valuation is the market value of its assets minus the valuation of its liability stream.1 The liability valuation is very sensitive to movements in interest rates. To the extent that plan assets include risky assets such as equities, asset valuations will be much less interest-sensitive than liability valuations, and so the plan’s funded balance (surplus) will vary in response to swings in interest rates as well as to swings in non-interest factors.

These statements can be expressed in terms of surplus variance, variance being the statistical measure of volatility. (The square root of variance is standard deviation.) For return on assets A and rate of change in liabilities L, the change in surplus is A-L, and the formula for surplus variance is:2

var(A-L) = var(A) + var(L) – 2*cov(A,L).

An interest-rate swap overlay accrues bond returns and pays cash returns. While this raises asset volatility, it can work to reduce surplus volatility by raising the correlation (covariance) between assets and liabilities (which affects surplus volatility negatively). However, when asset volatility is high and arises predominately from non-interest sources, then asset volatility will be the dominant contributor to surplus volatility, and the swap will do little to reduce surplus volatility.3

The volatility of equity returns is much higher than that of bond returns. For equity-heavy asset allocations—those with equity shares of 60% or more of assets—the non-interest-rate volatility of equities is the dominant contributor to surplus volatility, and so adding interest sensitivity via a swap will effect only a marginal decline in surplus volatility.

Furthermore, equities and other non-fixed-income assets do exhibit some sensitivity to—correlation with—interest rates. It is possible that allocating to the interest rate swap could make assets too sensitive to interest rates fluctuations. That is, after some point, additional allocation to the swap actually reduces the correlation between assets and liabilities, thus working to raise surplus volatility.4 The plan is more likely to make such an excessive allocation to the swap if it ignores the interest-sensitivity of its non-fixed-income assets.

Swaps and Duration

The discussion so far has described surplus volatility as depending on the volatilities of assets and liabilities and also their correlation with each other. Pension liabilities are generally very highly correlated with fixed-income assets, and so most people think in terms of duration when they consider asset-liability management. The discussion of the previous section can be recast in terms of duration, and this should make some of the points above more apparent.

Duration measures the interest-sensitivity of an asset. For example, a 30-year Treasury Strip changes in value by about 30% for every 100 bp shift in Treasury yields; its duration is 30 years. Other assets have higher or lower durations depending on whether their returns are more or less sensitive to interest swings.

It is a simple matter to calculate the duration of fixed-income assets, and it is equally simple to calculate the duration of pension liabilities, since they vary directly with interest-rate movements. Equities, however, are another story. Equity returns are historically correlated with returns on bonds, and this correlation is equivalent to duration.5 However, the correlation between equity and bond returns varies over time. The duration of equities can always be calculated in retrospect, but its prospective value is uncertain.

Exhibit 1 shows a moving ten-year-average of the duration of equities against that of long-term corporate bonds. On average over the 1960-2005 period, the duration of equities was 4.7 years, less than half that of long corporates. At times within this span, in the late-1970s and late-1990s, the duration of equities exceeded that of long bonds. At other times, in the late-1960s and again recently, it was sharply negative. (In those episodes, returns on equities declined sharply when interest rates fell, and vice versa.)

Exhibit 1
Durations of Bonds and Equities
Source: Ibbotson Associates, Western Asset

Because of this variability, some consider the duration of equities to be a nebulous concept. However, as argued below, the magnitude of the duration of equities directly affects the proper allocation to an interest-rate swap, so one has to consider it when using swaps as an LDI tool.

Pension liabilities typically have very long duration. An equity-heavy asset allocation usually has much shorter duration. This mismatch leaves pension surpluses susceptible to swings in interest rates. In effect, a typical pension surplus has negative duration: it declines in value when interest rates decline (and vice versa), because interest rate declines raise liability valuations more sharply than they do asset valuations. Furthermore, as the duration of equities fluctuates, surplus duration varies with it. The lower the duration of equities, the more negative surplus duration is.

A swap can reduce surplus volatility by increasing the duration of plan assets, thus raising surplus duration toward zero. The optimal allocation to the swap is that which matches the dollar-duration of assets to that of liabilities, thus achieving zero surplus duration. Any lower allocation to the swap, and the surplus would still decline when rates decline (still-negative surplus duration). Any higher allocation to the swap, and the surplus would rise opposite declining rates (positive surplus duration). Either type of fluctuation (duration) contributes to surplus volatility, and so the optimal allocation to the swap is that which eliminates interest-sensitivity.

Exhibit 2
The Optimal Swap Allocation Varies Over Time
Source: Ibbotson Associates, Western Asset

The dollar duration of (all) assets is relevant here, because interest rate swings affect valuations of all liabilities. To remove all duration from the surplus, the swap must match total asset duration—not just that of fixed-income assets—to that of the liabilities.

This is where swap overlays become tricky. Because the duration of equities varies so much over time, it is nearly impossible to quantify with any confidence, and so the optimal allocation to a swap is similarly uncertain, since it depends on the duration of equities. A swap allocation calibrated on (optimized according to) one particular historical episode and its attendant duration for equities could be very sub-optimal under different conditions. It could even work to raise surplus volatility. This would be so if the swap achieved a positive surplus duration larger in magnitude than the initial, negative surplus duration.

This could easily happen, given the short-term fluctuations in the duration of equities observed over recent decades. Exhibit 2 shows the optimal allocation to swaps over time, for a hypothetical pension plan allocated 60% to equities and 40% to long bonds, given the observed duration of equities over the preceding ten years (as shown in Exhibit 1). As seen in Exhibit 2, the 140% allocation to the swap that would have been optimal over 1996-2005 (when equities’ duration was negative) would have been grossly excessive—and counterproductive—over 1989-98, when equities’ duration matched that of long bonds. Other periods have shown comparable—though milder—such fluctuations in market conditions.

Even if the swap were to work perfectly, achieving zero surplus duration, the surplus will still be sensitive to non-interest variability. With heavy asset allocations to equities, that non-interest surplus volatility will be larger than the interest-sensitivity the swap addresses. The swap will not effect a critical reduction in surplus volatility, and what benefits it does achieve are tenuous, being dependent on the exact level of interest-sensitivity of equities during the period in which the swap overlay is put on.

Effective LDI: Simple is Better

Exhibit 3 illustrates these points, utilizing the risk/return experience of 1960-2005, for a hypothetical plan invested 60% in equities and 40% in intermediate-maturity fixed-income, with liabilities’ duration equal to that of long-term corporate bonds, 9.4 years. Over 1960-2005, equities exhibited an average duration of 4.7 years, and so they contribute 2.8 years (60% of 4.7) to the “dollar duration” of assets. Intermediate fixed-income showed duration of 5.4 years over that period, and so they contribute 2.2 years to the total dollar duration of assets.

To achieve zero surplus duration, the swap should contribute 4.4 years of dollar duration to assets, which would require a notional allocation to the swap equal to 47% of total assets.6 Were that allocation to work as intended, achieving zero surplus duration, the net reduction in surplus volatility would be 118 bps, from 10.53% per year to 9.35% per year.

Now, the 4.7 year duration used for equities in Exhibit 3 is an average across the 1960-2005 experience. As the duration of equities fluctuates in the short run either above or below its 1960-2005 average, the benefits of the 47% swap allocation will decline from their initially modest level. Surplus duration will be positive when the duration of equities is above its long-term average, and it will be negative when the duration of equities is below average. Either movement raises surplus volatility, thus eroding the risk-reduction benefits of the 47% swap allocation will decline from their initially modest level. Surplus duration will be positive when the duration of equities is above its long-term average, and it will be negative when the duration of equities is below average. Either movement raises surplus volatility, thus eroding the risk-reduction benefits of the 47% swap allocation.

Exhibit 3
The Optimal Swap Matches Asset and Liability Durations, But Achieves Slight Only Reduction in Surplus Volatility
Source: Ibbotson Associates, Western Asset

In contrast to the tenuous benefits of an optimal swap, a pension plan can take steps that are sure to produce some decline in surplus volatility. The simple expedient of extending duration on existing fixed-income allocations is virtually certain to reduce surplus volatility. Such a tactic would match the duration of fixed-income allocations to that of the liabilities. With such an allocation, the surplus would still generally exhibit negative duration, but less so than before, so that total surplus volatility would be smaller.7

The effects of these various tactics are depicted graphically in Exhibit 4.8 As seen there, a 60/40 equities/notes allocation results in substantial surplus volatility. Extending duration on fixed-income assets—replacing note exposure with long bonds—will almost surely produce some benefits. Further (optimal) allocation to a swap could yield slight additional benefits, but too high an allocation to the swap could result in higher surplus volatility than without the swap.

Note that all these attempts at LDI leave the pension allocation much closer to its initial profile than to that of an immunized (100% bond) allocation. With heavy allocations to equities, non-interest risk is the dominant contributor to surplus volatility, so addressing interest risks via LDI tactics—without reducing equity allocations—cannot make a critical difference in plan performance. Even the modest benefits the swap provides in the best of circumstances fade as the duration of equities varies above or below the expected levels upon which the swap is predicated.

Exhibit 4
Bond Swaps Effect Only Slight Reductions in Surplus Volatility
Source: Ibbotson Associates, Western Asset

Bond Swaps and Funding Ratios

Our results so far implicitly assume that the plan’s funded balance was close to 100% (near-zero surplus or deficit). It is the case that as funded status declines below 100%, swap overlays become more beneficial. This is so because the larger the deficit, the larger liabilities are relative to assets, and so the more important the interest-sensitivity of the liabilities becomes relative to the non-interest sensitivity of the assets. For extremely under-funded plans—funding ratios of 70%, 60%, or less—liabilities will dominate the benchmark, and swaps will be much more effective.

The converse is also true. As funded status rises above 100%, the benefits of swap overlays decline, because the importance of the interest-sensitivity of liabilities declines as assets rise relative to liabilities. For plans with funded status of 130% or higher and equity-heavy allocations, a swap overlay would effect hardly any decline in surplus volatility. Sustaining substantial pension surpluses is inevitably “hostage” to equity-market performance (unless the allocation is substantially immunized).

While the benefits of swaps are higher for heavily under-funded plans, the required allocations to swaps are commensurately higher as well, causing greater attendant complications (counterparty risk issues, swap fees, etc.) Even in these circumstances, while swap overlays provide more benefits, they still fail to effect a critical reduction in surplus volatility. That is, an interest-hedged equity-heavy allocation more closely resembles its unhedged “cousin” than it does an immunized allocation, regardless of funding levels.

Bond Swaps and Risky Assets Other Than Equities

Our results hold mostly because equities are risky, but partly because they are usually positively correlated with bonds. Suppose a pension plan were invested heavily in assets such as real estate, private equities, or non-U.S. equities. If the volatility of these assets is comparable to that of equities, then it will still be the case that putting on an optimal swap overlay to hedge interest-rate risks will achieve only slight reduction in surplus volatility.

If these assets are as highly correlated with bonds as are large-cap equities, then the chances of an over-allocation to a swap overlay are just as high as in the equity-heavy case discussed above, and optimal swap allocations will be substantially less than 100% of assets (liabilities). If these assets are less correlated with bonds than are equities, then the optimal swap will effect more of a reduction in surplus volatility than it would for equity-heavy allocations. However, even this larger reduction in surplus volatility will be less than critical, because asset volatility will still be the dominant contributor to surplus volatility, and even less of the asset volatility will be interest-sensitive (by assumption).


With pension reform imminent or in place here and abroad, corporate defined-benefit plans worldwide are considering efforts to reduce short-run surplus volatility. Swap overlays are an intuitively attractive strategy to this end. However, swap overlays address only the interest-sensitivity of assets versus liabilities, and so much of the volatility of equities is accounted for by non-interest factors that merely removing interest rate risks does little to reduce surplus risk for equity-heavy allocations. Swaps entail detailed effort for little gain.

Over-allocating to a swap can be counterproductive, leaving higher surplus volatility than otherwise, and the optimal swap allocation is very dependent on the assumed durations of non-fixed-income assets. As financial-market conditions vary, what was construed to be an optimal allocation to the swap given past conditions could easily become counterproductive, voiding even the modest gains that the swap ideally could have achieved.

Exhibit 5
What is Your Relevant Planning Perspective?
Source: Western Asset

The simple expedient of replacing existing intermediate-maturity fixed-income and cash asset allocations with long-duration bonds is a cleaner tactic. It also provides only modest reductions in surplus volatility, but these benefits are sure and simply achieved.

LDI tactics are intended to minimize the short-term volatility of the pension plan surplus. Pension plans invest heavily in equities in order to enjoy the long-run benefits equities afford through higher expected returns. The decision to pursue long-run benefits is a valid one, and we are not criticizing it. However, pursuing long-run benefits via the equity market leaves a plan susceptible to the short-term risks that will be more obvious and more onerous following FASB and ERISA reforms.

It is tempting to look for some magic bullet that removes short-term risks while leaving the long-term benefits of equities in place. Many plans are looking to custom swaps as such a cure-all. The analysis here indicates that these hopes are misplaced.

There is no effective tactic available that will leave a pension plan open to the long-run benefits of equities and other similarly risky investments while also shielding it from the accompanying short-term risks. Plan managers would do better to face the long-term/short-term trade-offs head on and decide accordingly, rather than pursuing costly and onerous swaps that won’t deliver their intended results.


Leibowitz, Martin (1986), “Total Portfolio Duration: A New Perspective on Asset Allocation,” Financial Analysts Journal, September-October 1986.

Fama, Eugene and Kenneth R. French (1993) “Common Risk Factors In The Returns On Stocks And Bonds,” Journal Of Financial Economics, vol. 33, pp. 3-56.


  1. That is, the stream of future pension obligations discounted back to an aggregate present value using high-grade corporate bond yields as a discount rate. This formula holds for FASB, IAS, and FRS accounting protocols as well as U.S. and U.K. government regulations.
  2. This formula holds as written for plans that are fully funded. If the plan’s funded balance, F, is different from 100%, then the change in surplus will be A-L/F, and the formula for surplus variance changes accordingly. We discuss below the effects of funded levels on our results.
  3. An alternate way to see this is to treat the swap, for the moment, as part of the liabilities. In effect, the swap transforms the liabilities from bond-like to cash-like. However, if asset volatility is much higher than that of bonds or cash—if var(A) >> var(L), then asset volatility will dominate surplus volatility regardless of whether or not the swap is in place, and so the swap has little impact.
  4. Another way to understand this point is by treating the swap as part of the liabilities, as in footnote 3. Again, the swap can be seen as converting the liabilities from long-duration to cash. As equities are typically more highly correlated with bonds than with cash, in this context, the swap can be seen to reduce the correlation between assets and liabilities, which works to raise surplus volatility as per the formula in the text.
  5. Typical definition and usage of duration concerns only movements in the levels of Treasury yields. For the purposes here, we define the duration of equities in terms of their total correlation with corporate bonds returns. That is, in line with Leibowitz (1986), the duration of equities is defined as DUR-EQ = eb•(e/b)•DUR-CORP where eb is the correlation between equities and corporate bonds, e is the standard deviation of equities, b is that of corporates, and DUR-CORP is the duration of corporates. This definition reflects the sensitivity of equities not only to movements in interest rates proper (Treasuries), but also to movements in corporate yield spreads. Since pension valuations under both FASB and ERISA procedures use corporate-yield discount rates, this sensitivity of equities to both sources of fluctuations in corporate yields is the valid metric for purposes of managing surplus volatility. Furthermore, Fama and French (1993), among others, find that both interest-rate and corporate spread swings have enduring explanatory power for equity returns. In other words, they find this sensitivity of equities to both yield and spread changes to be an enduring phenomenon. While our usage of duration differs from the norm, it is actually that most favorable to bond swaps. Using the standard, “interest-only” definition of duration in our analysis would deliver “optimal” swap allocations that were even less beneficial than what we find to be the case in the text.
  6. The swap is a zero-investment allocation. Since the pay side of the swap has zero duration, an allocation to the swap in this notional amount affects total asset and surplus duration the same as would a cash allocation to bonds equal to this share of assets.
  7. This would be the case as long the duration of equities were less than that of the liabilities. That is, in episodes where the duration of equities was greater than that of long bonds, such as in the

    late-1970s and late-1990s, extending duration on fixed-income allocations could conceivably push surplus duration to positive levels, in which case, surplus volatility might not be reduced on net. However, such episodes are extremely rare. Furthermore, note that in such an episode, while the benefits of extending duration would be reduced or removed, the benefits of a swap overlay would be even more sharply affected. Any substantial allocation to a swap overlay would be almost surely detrimental in such an episode, so even then, extending duration on fixed-income allocations would be more effective than allocating to a swap.
  8. The reader should not be swayed by the increases in expected return that swaps are depicted as achieving here. The point of the swap is to reduce surplus volatility, and the gains in the chart in that direction are slight or non-existent. The gains in expected returns depicted occur because long bonds far outperformed cash over the historical period on which the chart is based. That superior return could be expected to occur in the future only if bonds can again be expected to outperform cash. But if one really wanted to bet on that, a whole other set of strategies would be in order. Again, the lesson of this chart derives from the effects on surplus volatility, not from the apparent return boost.
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