# LDI and the Persistence of Tracking Error Appendix

#### Section 1. Introduction.

In the white paper *LDI and the Persistence of Tracking Error*, we report the results of various hedging exercises involving both simple and complex LDI strategies. We analyze how well these strategies track liability valuations that were derived from using yield curves to discount projected DB benefit cash flows to present values. We find that simple LDI strategies—involving optimized allocations to long government and long credit—provide nearly the same tracking error against liabilities as do complex LDI strategies involving either key-rate-duration-matching or cash-flow-matching allocations.

The present paper provides the details behind these results. Section 2 describes the process of estimating the yield curves used both to discount liabilities and also to estimate hypothetical Treasury STRIPS used for cash-flow-matching allocations. Section 3 describes the behavior of these yield curves across out historical sample 1988–2012. Section 4 describes the behavior of liability valuations derived from these yield curves. Section 5 provides the details of our hedging exercises.

#### Section 2. Estimating Yield Curves from Bond Market Data: Why Curves?

The reader might be wondering why we even utilize yield curves to evaluate liabilities rather than use a single, long AA yield, as corporate plans commonly do for GAAP reporting. Well, when a single yield is used to discount all cash flows to present values, then the duration of that valuation is indeed the sum of the contributions to duration of each cash flow, and one can group these contributions to determine how much duration there is in various maturity segments of the cash flows.

However, it would be wrong to describe these as key-rate durations, because each of these components of duration depends on the same long AA yield, and they depend not at all on the yields at different points along the yield curve.

Thus, there is no reason to believe that a key-rate-duration-matching or cash-flow-matching portfolio will closely track such liability valuations. Such complex asset allocations offer the advantage of being sensitive to movements in specific components of the yield curve, but the liability valuations display no such sensitivity. So, we introduce yield-curve-based liability valuations in order to provide the most hospitable environment for the complex LDI strategies. In the white paper itself, we report the results only of hedging exercises involving curve-based liability valuations. However, we also performed these exercises on liability valuations derived for use of a single long AA yield as a discount factor, and such results are discussed in Section 3 along with the results from the paper.

Another reason that the reader might be interested in these yield curves is that the PPA curves are indeed relevant for corporate plans’ reporting to the PBGC. Under federal law through 2011, corporate plans had the option of evaluating liabilities either by using the PPA curves discussed here or by using a two-year average of those curves, with the resulting yields further simplified down to three maturity buckets and those yields applied to cash flows falling within the corresponding maturity ranges. Legislation passed in the summer of 2012 changed the latter option from a two-year average to a 25-year average of the PPA curves. As the Treasury has not in fact yet constructed nor published such curves for years prior to 2003, our estimates of such curves reported here might be of interest in their own right, as a glimpse of what the Treasury’s curves should resemble.

**Estimating the Yield Curves.** The yield curves used to discount DB cash flows were derived using the Treasury’s methodology, which produces high-quality corporate bond yield curves in accordance with the Pension Protection Act of 2005. The Treasury’s methods are described in a number of papers, and slideshow presentations are available on its website. The most succinct and detailed such document is Girola’s (2007).

Can’t a corporate bond yield curve be derived immediately, straightforwardly from corporate yield data across various maturity bonds? The answer is no for a number of reasons. First, DB pensions’ estimated cash flows typically stretch 70 to 100 years into the future, so market-data yield curves extending only to 30 years must be extrapolated by some method. Second, yields to maturity on available bonds describe the return on the whole horizon of cash flows accruing to a particular bond, and yields equivalent to zero-coupon corporate bonds must be used to discount individual DB cash flows. Third, corporate bond yields will vary by credit quality as well as by maturity, so some method is needed to extract information from bonds of different qualities in order to “draw” a single yield curve representative of the whole universe of available issues.

The Treasury’s PPA yield curve provides spot yields (essentially zero-coupon corporate bond yields) extending out to 100 years maturity estimated by aggregating across the universe of A through AAA rated corporate bonds. Now, the Treasury’s official data on this curve currently extends back only to 2003, so in order to perform hedging exercises across a longer sample period, we need to replicate the Treasury’s process across years prior to 2003. Once again, Girola (2007) provides all the details behind this estimation process. We recount the most relevant details here.

Corporate bonds of rating A through AAA with maturities of 30 years or fewer were included in the universe for yield curve estimation. Bonds with explicit call or put options were excluded. Also, to anchor the yield curve at the short end, Federal Reserve data for yields on AA rated commercial paper of varying maturities was also included in the universe. While the Treasury’s estimation methods employed the universe of corporate bonds reported by Merrill Lynch, we utilized Barclays POINT data. Price, coupon, rating, and other relevant data for individual bonds are available on POINT back to 1988.

In estimating its yield curve, the Treasury assumes that the forward rate curve associated with the spot yield curve follows a cubic spline shape. Forward rates represent yields on essentially instantaneous (zero-maturity) instruments at different points in the future. Forward rates can be calculated from the par coupon curve, and vice versa. Forward rates can also be aggregated to determine spot yields.

The Treasury assumes that the cubic spline representing the forward rate curve is a B-spline with “knot points” at maturities of one-and-a-half, five, three, seven, 15, and 30 years. In other words, successive segments of the yield curve between these points will have slightly different shape properties from each other. The Treasury also imposes one endpoint constraint on the shape of this yield curve at the zero-maturity and two shape constraints on the curve over the 15- to 30-year maturity range. Finally, the Treasurey specifies quality adjustments that allow bonds of the same maturity but different rating to exhibit different prices (and different yields).

Upon defining δ(t) as the price of a zero-coupon corporate bond with maturity t, the market price of a given corporate bond P can be expressed as:

where C_{t} is the cash flow accruing to the bond at time t, M is the maturity of the bond, x_{1} and x_{2} are variables depending on the rating of the bond and its duration, and α and β are parameters to be estimated; x_{1} differentiates the price of AA bonds from those of AA binds with the same maturity; x_{2} differentiates AAA and AA bonds from A bonds with the same maturity.

Given the cubic spline shape and endpoint constraints the Treasury imposes, each δ(t) can be expressed as a non-linear function of the same five spline parameters. So, the equation above is a non-linear form involving seven parameters in total. For a given sample of bonds, the Treasury estimates these parameters via weighted non-linear least squares regression. For each commercial paper issue, (squared) weights are inversely proportional to the number of commercial paper issues utilized. For couponed corporate bonds, (squared) weights are proportional to the outstanding par value of the bond and inversely proportional to the bond’s duration.

The Treasury estimates yield curves for each trading day of a month, and then the relevant curve for that month is a simple average of the curves for all trading days in the month. In our replication of the PPA curves, **we estimated curves only for the last trading day of December of each year**. Other than this simplification and our use of Barclays POINT data, our construction of PPA replicate yield curves followed every detail of the Treasury’s method. Once again, our PPA curve replicates were estimated for December 1988 through December 2012.

Since the Treasury’s PPA curves are estimated from the universes of all bonds rated A through AAA, these curves are not usable for evaluating liabilities for GAAP financial reporting, as FASB’s protocols specifying “high-grade corporate securities” are generally interpreted as meaning AA or higher. While it might seem that the average rating of an A to AAA universe is AA, in fact, A bonds comprise 90% percent of the A to AAA universe presently and at least 80% of the universe across the 1988–2012 sample, so the average rating for this universe is more like A/A+.

In order to provide GAAP-compliant liability returns, we estimated a series of yield curves that followed all the spline specifications outlined here and utilizing the same universe of bonds, but restricting the estimation to AA bonds only. That is, equation (1) was estimated with the same spline specifications, with the quality terms x_{1} and x_{2} suppressed, and with only AA bonds included in the run.

Finally, when it comes to producing portfolios of STRIPS that cash-flow-match DB liabilities, there is the problem that DB cash flows typically extend out 70 to 100 years, while STRIPS are available in the market only to maturities of at most 30 years. Using the longest-available STRIPS to hedge all cash flows from that maturity to 70/100 years raises the issue of whether those out-year flows are properly matched.

To deal with this problem in a “laboratory setting,” we estimated yield curves for Treasury STRIPS that followed the same spline specifications as discussed above. In effect, equation (1) was estimated with the quality terms x_{1} and x_{2} suppressed and utilizing data only for Treasury STRIPS. Yield curves in successive years were then used to calculate prices and returns on hypothetical STRIPS existing at each maturity point out to 100 years. These hypothetical STRIPS were then used to cash-flow-match the cash flows comprising the liabilities. Our expectation is that since these STRIPS curves exhibit exactly the same shape properties as the yield curves utilized to discount DB cash flows, these hypothetical STRIPS portfolios have the best possible chance of tracking the fluctuations in liability valuations as yields change.

#### Section 3: The Historical Behavior of Our Yield Curves.

Once again, the historical performance of the yield curves we have estimated should be of interest. They offer a look at how the Treasury’s own PPA curve should behave when the Treasury does publish PPA yield curves going back 25 years as required by last summer’s legislation. The movements in these curves relative to movements in long AA yields also provide some understanding of how curve-based liability valuations behave differently from typical corporate DB liability valuations based on a single Long AA yield. Finally, observing the disparate movement of our PPA and AA curves versus those of our STRIPS curves provides some understanding of how hard it is to properly hedge curve-based liability returns using even the most targeted STRIPS portfolios.

So, notice first in Exhibit 1 how the Treasury’s PPA curves behaved across the financial crisis of 2007 to 2009. Keep in mind that long AA spreads were relatively stable across the financial crisis, so that when long Treasury yields plunged in 2008, long AA yields dropped as well, from 6.26% to 5.61%, and then when long Treasuries sold off in 2009, long AA yields rose back to 5.80%. In contrast, the PPA curve rose sharply in 2008 at maturities out to 26 years, thanks to a near-collapse in the commercial paper market, and only at longer maturities did spot yields fall. Then, when commercial paper markets recovered and spreads contracted in 2009, the PPA curve showed sharp declines in yields out to 30-year maturities.

This behavior results in much different behavior of liability returns when our yield curves are used from what occurs when a long AA yield is used. As we’ll see in the next section, applying the long AA yield as a discount factor to a standard set of DB cash flows results in a 14.1% liability return in 2008 and a 3.3% return in 2009. In contrast, applying the PPA curve to those same cash flows results in liability returns of 0.0% in 2008 and 18.2% in 2009. Using our replicate of the PPA curve results in liability returns of 2.5% in 2008 and 13.6% in 2009.

Obviously, the choice of discount method has important implications for pension performance during financial crises. As we will see, at other times the differences are less dramatic.

Exhibit 2 shows our replications of the PPA yield curve across the crisis. The broad contours of the respective sets of curves are similar, but there are some differences, as seen more immediately in Exhibit 3. Once again, our estimation techniques followed exactly the same shape constraints and quality adjustment as the Treasury specified. Our process differed only in using and parsing the Barclays POINT dataset rather than Merrill Lynch’s dataset and in estimating curves on the last trading day of each December rather than taking an average of curves across all the trading days of each December.

In the case of the December 2008 curves, the differences between the actual PPA curve and our replicate of it most probably reflect our use of last-trading-day data. Corporate yields were declining across that December, especially so in the commercial paper market, so averaging the December data would seem to account for the slight degree to which the actual PPA curve lies above our replicate. Certainly, the shape of these curves is strikingly similar, and as seen above, the liability return pattern for the two curves are very close.

Exhibits 4 and 5 show our replicates of the PPA curve over the last three years as well as across the financial crisis years of 1998 to 2001. It is clear from these respective charts that market behavior has quieted down since 2009 (not a big surprise) and that yield volatility was much worse across the recent financial crisis than was the case during the crises evoked by Long-Term Capital (1999), 9/11, and Enron (2002).

Notice that even in the midst of the financial crisis in 2000 and 2001, the PPA curves did not display a hump comparable with that in the curve for December 2008. The curve shifted up in mostly parallel fashion in 1999, flattened in 2000 with short rates rising (when the stock market began its selloff), and dropped sharply at the short and long ends in 2001, when recession set in and when terrorist attacks occurred. Rates then dropped in mostly parallel fashion in 2002, but at no time did short rates spike up and long rates plunge such as occurred in late 2008.

As mentioned above, the estimations of equation (1) for PPA curves involving the A to AAA bond universe result in price premia being allotted to AA bonds over A bonds of comparable duration (as well as for AAA bonds over AA bonds). These quality premia are substantial and statistically significant across the 25 years for which we have estimated yield curves using this methodology. For example, for the December 2012 estimation, the estimated equation allots a price premium to AA bonds over A bonds of 0.266 price points per year of duration. So according to this curve, an AA bond with 12 years duration should sell for nearly 31/4 points more than an A bond with the same duration.

One would expect that this quality premium would result in uniformly lower spot yields for the AA curve than for the (A to AAA) PPA curve, but this is not the case. As seen in Exhibit 6, for December 2012, the AA only and PPA replicate curves cross at the 22-year maturity point, with A to AAA yields lower than AA yields past that maturity point. Similar behavior is exhibited for other years in the sample.

While some reasons for this phenomenon can be conjectured from the properties of the estimation processes, the lower level of the PPA spot curve at longer maturities may in fact be consistent with market fundamentals. Even though PPA curve spot yields are lower than AA curve **spot** yields past 22 years maturity, the **par coupon** yield curves bootstrapped from these conform with expectations, as seen in Exhibit 7. Par coupon yields are nowhere higher for AA bonds than for the A to AAA bonds covered by the PPA curve for December 2012 data. A higher price for higher-rated bonds implies only a lower yield-to-maturity (par coupon yield). It does not necessarily imply lower spot yields everywhere, and the spot yields shown in Exhibit 6 are in fact consistent with this weaker property of market pricing by quality.

Still, our findings are that PPA-conformable estimation processes can result in an A to AAA curve that lies below the corresponding AA only curve over wide ranges of maturities, and it is quite possible that using the PPA curve would result in a higher liability valuation than if a plan used the AA curve or Long AA yields. So pension plans should be aware that choosing the “lower-rated” A to AAA curve to discount its liabilities may not provide a lower liability valuation and higher funded status. Meanwhile, choosing a curve instead of a single Long AA yield could also provide a higher liability valuation, because of the lower yields used to discount shorter-maturity cash flows (given a positively-sloped spot curve).

A final observation drawn from the estimated historical yield curves has to do with spreads. When one uses STRIPS to key-rate-duration-match DB liabilities, those STRIPS will closely track the movement in liability valuations if corporate spot yields and corresponding STRIP yields move by the same amount, in other words, if (spot) spreads are constant. So, the more stable the spread curve as yields change over time, the better will STRIP be able to hedge the liability valuations and vice versa.

By the same logic, when spreads do change, a broad credit index will be better able to hedge this movement the more the spread curve moves in parallel fashion. If the spread curve “twists,” with spreads at different maturities changing by different amounts, then something akin to corporate STRIPS will be needed to hedge such disparate movements.

In sum, the degree of movement in the spot spreads curve over time will be an indication of how well STRIPS can hedge DB valuations, and the degree of twists and turns in the spread curve (as opposed to parallel movements) will be an indication of how well simple spread-duration-matching tactics might work. Exhibit 8 shows spread curve over recent years for the PPA curve relative to the STRIP curve, while Exhibit 9 shows the same spreads for the AA curves over STRIPS curves.

It is obvious from these charts that both absolute movements in spreads and twists in the spread curve are substantial from year to year. For spreads off the PPA curve, there are substantial swings in the spread curves from year to year for every year except between December 2009 and December 2010. For spreads off the AA curve, the results are similar. Notice the extremely wild swings in spread curves from December 2007 to December 2008 to December 2009. Notice that also while spreads for the Barclays Long AA Index changed only modestly between December 2007 and December 2008 (from 178 bps to 277 bps), spot spreads for the AA curve swung especially dramatically then, with short-dated spreads rising by as much as 325 bps and long-dated spreads **dropping** by 150 bps. With spread curves swinging so wildly across our sample, it is not surprising that key-rate-duration-matched STRIPS portfolios fail to closely hedge liability returns or that broad allocations to credit are of only modest effectiveness in hedging spread risk.

#### Section 4. Using Spot Yield Curves to Generate Liability Returns and STRIP Assets To 100 Years Maturity.

The series of PPA and AA curves were applied to standard sets of DB cash flows to generate series of liability returns. We used two sets of cash flows shared with us by corporate clients, one with somewhat longer duration than the other. We’ll report here on the results for the longer-duration set of cash flows.

Yield curves at the start of each year were used to discount the cash flows to a present valuation. Actually, the yield curve for the last trading day of the previous year was used. Then, the cash flows rolled down one year in maturity, and the yield curve for the last trading day of the year was used to discount those matured cash flows to a present value. This process mimics both the changes in yields and the maturation of liabilities that occurs during a year. The percent difference between these two valuations was then recorded as the liability return for that year for that curve.

While our liability returns can be thought of as a time series, they do not show the evolution of a plan’s cash flows over time. This is because the set of cash flows are held static. They are allowed to roll forward one year for determining the end-of-year valuation, but then for the following year, the cash flows are re-initialized and the process starts anew. In effect, the set of liability returns represents a distribution of liability returns corresponding to (deriving from) the distribution of yield curve changes embedded in market history from 1988 to 2012. We’ll draw upon this point in the next section.

For purposes of comparison, we also constructed a set of liability returns drawn from the same cash flows, but using the yield-to-worst for the Barclays Long AA Corporate Index as a discount rate rather than the yield curves already described.

Exhibit 10 shows annual liability returns over 1989 to 2012 using our PPA curve replicate, our AA curve, and the Long AA yield as discount tools. As seen there, the three measures adhere pretty closely over 1989 to 1999, but there has been more divergence since then, especially over the 2007 to 2009 financial crisis. The AA curve and Long AA valuations exhibit very different behavior over 2008 to 2009 from what the PPA replicate curve does.

It is interesting to note that the curve-based valuations show slightly higher average returns than the long AA based valuation. It is also the case that the curve-based valuation show lower volatility. We can offer two reasons for this. First, a curve would seem to show less volatility in aggregate than would any one point on the curve, hence less volatility in a curve-based valuation. Second, since the spot yield curves used are generally positively sloped, the lower yields at the short end of the maturity range result in short-dated cash flows accounting for a greater share of the total liability valuation than would be the case were only a single long yield used as a discount rate. This means that the curve-based valuations will generally feature less duration and, therefore, less volatility, since duration is merely the weighted average of the maturities of the cash flows, and positively-sloped curves result in higher weights for shorter-maturity cash flows.

Lower duration would also seem to account for the much lower volatility of the PPA curve liability returns than the AA curve returns. However, as seen in Exhibit 11, the PPA curve valuations show only slightly lower duration than the AA curves, hardly enough to account for a 100 bps per year difference in volatility. Instead, it would seem that the PPA curves themselves are indeed less volatile than the AA curves, despite the fact that AA spreads seem to be less volatile than spreads of lower-rated bonds.

#### Working With STRIPS.

In this study, Treasury STRIPS were the main tool for fine-tuning yield curve matches within complex LDI matching strategies, and regression analyses were the main tool for determining optimal hedges. With a 24-year sample period (distribution) for liability returns, we cannot key-rate-duration match every point on the maturity range, as the number of maturities far outnumbers the size of our sample. So regressions must take a more targeted approach.

A natural set of targets are the segments of the yield curve determined by the Treasury’s estimation method. As mentioned above, the cubic spline through which Treasury estimates its yield curves features “knot points” at maturities of one, three, five, seven,15, and 30 years. This naturally splits the yield curve into segments of 0 to one-and-a-half one-and-a-half to three, three to seven, seven to 15, and 30-up years; each segment will display somewhat different properties for the resulting yield curve. It is logical to construct a suite of STRIPS, one targeting each segment.

A detail to keep in mind is that for hedging the yearly liability returns, the relevant yield curve is that at year-end that will be used to discount end-of-year cash flows. As just discussed, that yield curve will be applied to the cash flows after they have rolled down a year, to simulate the process of maturation of cash flows that occurs in a year. So, for example, the three-to-seven-year segment of the cash flows that will be discounted at year-end by spot yields along the three-to-seven-year segment of the curve actually starts the year with maturities of four-to-eight years, and similarly for the other segments. The trip targeted to this segment should have maturity approximately equal to the midpoint of the three-to-seven-year maturity range at year-end, so it should have one more year of maturity than that at the start of the year.

With this consideration in mind, we chose the following constant-maturity STRIPS to target the segments of the yield curve. The zero-to-one-and-a-half-year segment was targeted by a STIRP with beginning-of-year (BoY) maturity of one year, four and one-half months (maturing on May 15 of the year following the year in question). This STRIP has maturity of four-and-one-half months at year-end. The one-and-a-half-to-three-year segment was targeted by a STRIP with BoY maturity of three years, four-and-one-half months. The three-to-seven-year segment was targeted by a STRIP with BoY maturity of six years four-and-one-half months. The seven-to-15-year maturity range was targeted by a STRIP with BoY maturity of 12 years four-and-one-half months. The 15-to-30-year maturity range was targeted by a STRIP with BoY maturity of 23 years four-and-one-half months. As this last maturity was the longest for which a STRIP was constantly traded across the 1989 to 2012 sample, it was also chosen to target the 30-up year maturity range. Since the Treasury did not issue 30-year bonds in the late-1990s and early-2000s, the maturity of the longest available STRIP declined steadily over that period.

Finally, as already stated, the STRIPS yield curves we generated were utilized to construct a set of hypothetical constant-maturity STRIPS with maturities ranging from one to 100 years. At the beginning of each year of the sample, the previous trading day’s yield curve was used to determine the price of hypothetical STRIPS at each such maturity point, and then the subsequent end-of-year curve was used to determine the yield and price—and thus total return—of that STRIP over the course of the year (with the maturity rolled down by one year at year-end). These hypothetical STRIPS were used for the cash-flow-matching exercises.

#### Section 5. Results of Hedging Exercises.

**Constructing the Regressions.** When an efficient frontier is constructed within an LDI setting, optimal portfolios on that frontier are determined by manipulating the joint distribution of the liabilities and assets. As shown in Bazdarich (2006), the minimum-risk point on that frontier is the best available hedge of the liabilities, and it is equivalent to a regression (projection) of the liability returns on the asset returns.

As discussed in Section 4, our samples of liability returns represent the distribution of liability valuations, and the variance-covariance matrix of the asset returns and their vector of covariances with the liability returns describe the joint distributions of the assets and the liabilities. So, the portfolio weights determined by regressions of the liability returns on available asset returns are equivalent to identifying the minimum-risk point on an efficient frontier given the liabilities and relevant set of assets. In other words, these regressions will determine the best possible hedge of the liabilities that can be constructed from a given set of assets across the distribution represented by our sample period.

In performing these regressions, we imposed the same constraints on the portfolio weights that would be imposed in a portfolio optimization. That is, the weights for “cash” assets were constrained to sum to one, using Wald constraints. When overlays were allowed within the regression, the allocations to these were not included in the sum-to-one constraint, since these overlays were long/short allocations that did not use up cash assets.

Also, in these regressions, the constant term was suppressed, since it did correspond to return on any asset. It is a standard econometric fact that when the constant term is dropped from a regression, there is no presumption that the average of the fitted values will equal the average of the dependent variable. In the present environment, this means that there is no presumption that the optimal hedge will have an average return equal to that of the liabilities.

In the hedging results reported here, we list the average outperformance (alpha) of assets over liabilities as well as the standard deviation of this difference (tracking error). When this average outperformance is negative, the hedged portfolio underperforms the liabilities on average: funded balance would decline over time. As we will see shortly, in fact, the optimized hedges resulting from our study generally do underperform the liabilities.

The reader should keep in mind here that the least squares calculation still minimizes the sum of squared residuals (alphas), though there is no presumption that these alphas average to zero. Since the hedge determined by the regression is equivalent to the minimum-risk point on an efficient frontier, it can be inferred that no other portfolio (based on that particular set of assets) will deliver lower tracking error nor the same tracking error with higher average return. However, there may be some combinations that deliver higher average returns and only slightly higher tracking errors.

**Constructing Key-Rate Duration Matches.** Since our regression analyses are equivalent to optimizing a hedge based on the properties of the joint distribution of assets and liabilities, they are optimal hedges under the typical assumptions underlying such a joint distribution: namely, that each observation in the sample is an independent drawing from the distribution. When the actual data exhibit a trend, this runs counter to the assumption that each observation is independent of the others.

As discussed in this white paper, the liability returns show how the valuation of a static set of cash flows would have been affected by the interest rate behavior seen in various years. The constant-maturity STRIPS show how the returns on a static set of STRIPS would have been affected by that behavior. The returns on long credit, long governments, and the long G/C show how these indices were affected by that behavior. However, since these indices change over time, the change in index composition also contributes to performance. Also, since yield levels exhibit a downward trend across the 1989 to 2012 sample, this serves to lengthen the duration of both the liabilities and the fixed-income assets across the sample, but the STRIPS allocations feature essentially constant durations.

This raises a question about the optimality of our results. The trend toward lower yields over time may bias our regression results in favor of the long duration indices, since they share a common trend in duration with the liability returns that STRIPS do not. The key-rate-duration-matched portfolios were an attempt to offset this possible bias.

At each observation, end-of-year key-rate-durations were estimated for liabilities and STRIPS, and key-rate-duration-matched portfolios were constructed by allocating sufficient exposure to each STRIP to match the dollar duration of the corresponding segment of the cash flows. These constructs differ from the regressions involving STRIPS in that for the KRD-matched portfolios, allocations to each STRIP can vary at each observation, whereas for the regressions, optimal allocations to each STRIP are fixed across the sample. For each observation, resulting portfolio weights were summed across the range of STRIPS (yield curve segments), and then the difference between the sum of weights and one was financed/invested at LIBOR^{1}.

While the KRD-matched STRIPS hedge interest-rate risks as much as possible, they do not address spread risks, and the KRD-matched STRIPS have used up all essentially available assets. So, allocations to the credit overly were introduced here, just as in the regressions involving STRIPS and in the cash-flow-matching portfolios. In the regressions, optimal exposure to this overlay could be determined via the least squares process, but there is no analog of that here. Instead, allocations to the credit overlay were determined in analogous fashion to the determination of allocations to the STRIPS.

At each observation, the spread duration of the liabilities was calculated and sufficient exposure to the long credit overlay was chosen to match this spread duration. These allocations changed across the sample as the durations of liabilities and credit changed.

**Cash-Flow-Matching Portfolios.** Once again, the key-rate-duration-matching portfolios were intended to fully match yield-curve sensitivities of assets and liabilities. A different approach in the same spirit is to exactly match STRIPS to the DB cash flows. This serves to defease the cash flows, and the valuations of cash flows and STRIPS will differ only due to the different spot curves employed for each.

Now, a portfolio of cash-flow-matching STRIPS will typically cost substantially more than the corresponding liability valuation, because the Treasury spot yields used to evaluate the STRIPS will typically be well below the corporate spot yields used to evaluate the liabilities. Thus, it takes more than a dollar’s worth of STRIPS to match (defease) a dollar’s worth of liabilities. As with other applications of leverage in this study, the borrowing necessary to fully fund this allocation was assumed to occur at LIBOR.

Since the cash flows are static across observations, the portfolio of cash-flow-matching STRIPS is static as well. However, the market value of that portfolio changes as yield levels change, and its relative value against the liability valuation changes as well, so that the necessary amount of borrowing varies across the sample.

Finally, as with the key-rate duration-matching allocations, cash-flow-matching allocations use up more than all assets hedging interest-rate risks, leaving no assets available to hedge spread risks. As with the key-rate-duration-matching allocations, sufficient exposure to the long credit overlay was allowed so as to match spread durations of assets and liabilities. In fact, the notional exposures to this overlay were identical between the key-rate-duration-matching and cash-flow-matching allocations, since the choice between key-rate-duration-matching and cash-flow-matching has no effect on spread duration.

As noted in the text, allocations to this overlay were obviously unavailable to historically real-world investors. We employed these and other “hypothetical” instruments in order to determine how well liability risks could be hedged given the best conceivable instruments with which to hedge those risks. The fact that the simple, real-world LDI strategies perform nearly as well as these fanciful allocations is powerful evidence in favor of the simple strategies.

**Hedging AA Curve Liabilities With Long AA Credit And Hedging Long AA Valuations.** Two more features have been added to the results here that were not reported in this white paper. First, for the AA curve valuations, we also allowed for spread risks to be hedged via allocation to long AA corporate credit, rather than merely the long credit hedge that is used elsewhere. As reported in this white paper, long credit fares less well as a hedging instrument against the AA curve liabilities than it does against the PPA curve set, and our thought was that better results might be achieved by using a credit instrument that more closely matched the credit quality of the AA curve liabilities. We’ll discuss below whether long AA allocations are suitable for a real-world plan, but for now, it is at least worth a look to see how they perform.

Second, we also performed simple and complex hedging exercises involving liability valuations derived from the use of a single long AA yield as a discount factor rather than a yield curve. We argued at the outset that curve-based valuations provide a more hospitable environment for complex strategies than do valuations derived from a single long yield, and the results for the single-yield valuations provide perspective on this assertion.

**Statistical Results.** Exhibit 12 lists all these results. The results for the PPA curve based liability returns and for the AA-curve returns hedged by long credit are the same as those listed in the white paper. The only additions are the average errors (underperformances) shown in the leftmost column. As seen there, all our hedges tend to underperform the liabilities. Only when there is substantial leverage do the average portfolio returns come even close to those on the liabilities.

This speaks to a point discussed in the white paper, namely that both passive long government and long credit mandates will tend to fall behind liability returns over time—governments because they can’t match the corporate yield sported by liabilities, and corporates because periodic downgrades and defaults will pull the realized return on corporates below their “promised” yield. As mentioned in the text, this will be the case so long as these tendencies are not offset by either declining Treasury yields and/or widening spreads (since governments have longer duration and no spread duration). Well, in the sample period under discussion here, Treasury yields fell by 650 bps from start to end, and credit spreads widened by 113 bps, yet unleveraged long government portfolios still underperformed liabilities, and ditto for corporate.

Otherwise, the dominant result here is that simple strategies involving optimized allocations to long government and long credit perform as well or nearly as well as do the highly complex LDI strategies involving STRIPS. These results hold just as much for allocating utilizing long AA corporate credit to hedge AA curve liabilities, runs AA1 through AA10. There as well, the simple strategies deliver tracking errors little different from those for complex LDI strategies.

Interestingly enough, the results for the long AA liabilities, runs C1 through C10, show much the same pattern as for the curve-based liability returns. As asserted at the outset, the curve-based liability returns should be more hospitable, because they feature the key-rate durations that the STRIPS can target. The failure of complex strategies to significantly outperform the simple ones indicates that matching key-rate durations is not too beneficial. However, the fact that the complex strategies fare no worse against long-yield-based valuations than against curve-based valuations points to the benefits of extra duration per se, especially in a sample period where long rates dropped substantially.

As discussed above, we constructed year-by-year-optimized complex strategies in order to counter possible bias in our regressions. The results in Exhibit 12 suggest that the effects of any such bias are minimal. The year-by-year-optimized complex strategies do not perform as well as complex strategies determined by regression analysis, with portfolio weights fixed across the sample (and with that possible bias in force).

Finally, in this white paper, we argued that hedging spread risk is not just a matter of matching spread durations. Credit quality matters as well (and there is the related issue discussed in the white paper that attempts to reduce spread risk mismatch inevitably increase credit risk mismatch, since liability returns don’t exhibit credit risk, while corporate bonds do). The better results for long AA credit than long credit when hedging AA curve valuations provides further evidence to this point.

The question then arises whether allocations to long AA corporate credit are suitable for real-world plans. The Long AA Corporate Index currently contains only 82 securities, many of which are seasoned and trade infrequently. On paper, this index performs well, but real-world investors might have a hard time duplicating that performance. Another consideration with respect to the Index is that five years ago, it was dominated by financial issues, many of which suffered downgrades and defaults during the financial crisis. While these issues are gone from the Index, analogous problems could one day affect the bonds in the Index currently. With a small number of issues in an index, prospective credit risks become even more important.

The results for long AA credit versus long credit highlight how problematic the issue of hedging spread risk is. Readily tradable indices do not hedge this risk as well as do indices that are not so readily tradable. An alternative approach, as suggested in the white paper, is to stick to simple LDI strategies and seek to match average liability returns via active management.

#### References

Bazdarich, Michael J., (2006) “Separability and Pension Optimization,” *Journal of Fixed Income*, Winter 2006, pp. 1-8.

Girola, James A., (2007) *The Corporate Bond Yield Curve For The Pension Protection Act*, U.S. Department of the Treasury, October 11, 2007.

- Returns on LIBOR were proxied as the difference in total returns of funded and unfunded 30-year Swaps. This difference is the funding cost of a 30-year Swap and so is the relevant borrowing cost for leverage in the long-duration space.