The Impact of LDI on Portfolio Selection

Western Asset

I. Introduction and Summary

You are the manager of a pension fund, and you have been holding off incorporating active liability-driven investing (LDI) techniques because of a belief that the economic times are not right. What is the best time to move to an LDI approach, and just how sensitive are the benefits of an LDI framework to differences in the financial-market climate? We address these issues in the present report. Specifically, we analyze the impact on portfolio allocation of a shift to LDI techniques across a variety of historical experiences.

We find that the impact of LDI techniques on optimal portfolio selection varies little across most financial-market environments. Regardless of whether we are entering an environment of falling inflation, rising and falling inflation, a golden age for bonds, or one resembling the summary experience of the last eighty years, shifting from a traditional (asset or total-return) framework to an LDI framework results in a sharp increase in long-bond exposure and a reduction in equity exposure. These results hold regardless of how aggressive your return target is.

These results are not uniform. Under market environments of secularly accelerating inflation or of the last ten years—bonds’ ”golden age,” optimal allocations within an LDI framework feature low exposure in long bonds. Meanwhile, under market environments of accelerating inflation or of a golden age for stocks (the 1950s), LDI allocations feature equity exposure comparable to the standard 60% prescription. However, optimal LDI allocations are consistent across the range of more normal market environments, and they typically feature the heavy long-bond and light equity exposures described here.

It is commonly asserted that consideration of liabilities “abandons [an objective of] return maximization.”1 We emphasize instead that LDI merely balances a different risk concept—surplus- rather than asset-risk—against the competing goal of maximum returns. We describe these techniques as “active LDI,” because while liability-hedging informs the investment process, the pension plan needn’t resign itself to passive (hedged) returns. It can seek aggressive return targets while still minimizing surplus risk relative to that target.

We can construct compact empirical assessments of the impact of LDI on portfolio allocation because of an analytical result derived from portfolio theory. This result states that within a surplus framework, all optimal portfolios can be decomposed into the liability-hedge and some level of exposure in an overlay portfolio that is independent of the liabilities. This result implies, in turn, that the impact on asset allocation of a shift to an LDI framework is independent of your risk tolerance or of the aggressiveness of your asset allocation, and we utilize this fact to generate the historic assessments described above. Another implication of this “decomposition property” is that optimal pension allocation is identical in practice to portable alpha management.2

II. Decomposing Surplus-Optimal Portfolios into a Hedge and an Optimal Overlay

In Western Asset (2006b), we described surplus-optimal portfolios as maximizing the expected change in—or return on—surplus for a given level of surplus risk. The expected change in surplus is merely the total return on assets less the change in liabilities. Since the rate of change in liabilities is unaffected by asset decisions, maximizing “surplus return” is equivalent to maximizing total return, the same as in a traditional, “total return” optimization framework.

Again, there is nothing in the LDI framework that requires a plan manager to give up total return. The LDI framework merely substitutes a more relevant risk measure against which to trade off higher expected returns. By expressing the surplus-optimal efficient frontier in terms of total return and surplus risk, we can more readily compare portfolios on that frontier with those on the “total-return” frontier.

In view of this fact, the terminology we have been using is deficient. Both surplus and “total-return” optimization can be seen to maximize expected total return relative to their respective risk measures, and so it might be confusing to use the term “total return optimization” to describe only the traditional framework. Still, to preserve comparability with our previous analysis, we will keep this designation. Keep in mind that what we call “total return optimization” refers to the traditional portfolio-optimization process of maximizing total return on assets for given levels of asset risk. What we call “surplus optimization” refers to maximizing total return on assets for given levels of surplus risk.

Exhibit 1 shows the efficient frontiers for surplus- and total-return-optimization, both measured against expected total returns targets. These are the same frontiers as we described in Western Asset (2006b), the innovation here being that the surplus-optimal frontier is now expressed in terms of expected total return rather than expected surplus return.3 In the next section, we compare portfolios along these frontiers, both for the 1960-2005 sample period underlying Exhibit 1 and for a range of other episodes.

Exhibit 1
Comparing Efficient Frontiers for Total-Return and Surplus Optimization
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset

A central result enabling that comparison is that portfolios on both frontiers can be expressed as—or decomposed into—two base portfolios: a minimum-variance portfolio and an overlay (zero-total-weight) portfolio that has maximum Sharpe Ratio. This result is known in finance theory as the Mutual Fund Theorem, and the details behind it are discussed in the Appendix. For now, the intriguing feature of this result is that on BOTH frontiers, incremental risks are taken (target returns are increased) via incremental positions in the same overlay.

This means that even though total-return-optimal and surplus-optimal portfolios maximize expected return relative to different risk measures, they differ only in their “starting value” portfolios. Adding an extra percentage point of target return to total-return- and surplus-optimal portfolios means adding exactly the same positions to each. So the difference in composition of surplus- and total-return-optimal portfolios with a 9% target return is exactly the same as the difference between surplus- and total-return-optimal portfolios with a 6% target, and so on. Describing how surplus- and total-return-optimal portfolios differ at any particular reference point necessarily also describes the impact of extending a surplus-optimization framework onto any total-return-optimal portfolio.

Efficient frontiers here are based on 1960-2005 asset returns and 2% per year non-systematic risk in liabilities.

The portfolio on the surplus frontier with minimum surplus variance is labelled MSVP. That on the total-return frontier with the same expected return is REP.

Other points on the frontiers show how those points can be constructed as combinations of either the MSVP or REP and appropriate exposure in the optimal overlay portfolio z*.

Exhibit 2
Comparison of Portfolios on Surplus- and Total-Return-Optimal Efficient Frontiers
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset

An obvious reference point for surplus-optimal allocations is the fully-hedged or minimum surplus-variance portfolio (MSVP). For most PBO liability valuations, one can closely approximate the MSVP via a 100% long-bond allocation. The total-return-optimal portfolio corresponding to this reference point is that which replicates the expected return of long bonds. By constructing this “replicate” portfolio under various historical regimes and comparing it to a 100% long bond allocation, we can measure the impact of LDI on portfolio allocation in each historical regime, regardless of the returns target chosen.

We are not saying that this replicate portfolio is desirable in any sense. We look at it only because it is a handy reference point relative to the presumably all-bonds MSVP and because the difference between total-return- and surplus-optimal allocations for this target is exactly the same as for any other target.

Exhibit 1 illustrates this point. The surplus-optimal frontier in the chart shows the MSVP portfolio, as well as portfolios corresponding to MSVP plus various levels of exposure in the optimal overlay portfolio, which we designate as z*. Similarly, the total-return-optimal portfolio with the same target return as that of the MSVP is designated as REP (for replicate), and we identify points on the total-return frontier that correspond to REP plus various levels of z*. As the chart indicates, regardless of the level of target return, the difference between the surplus- and total-return-optimal portfolios is the portfolio weights MSVP-REP. By identifying these differences MSVP-REP, we identify THE impact of LDI on portfolio selection.

III. Optimal Surplus Portfolio Allocation Across Various Historical Episodes

In order to analyze the impact of LDI under a range of circumstances, we specify six disparate periods within the eighty years of returns experience reported by Ibbotson and Associates:

  1. the long run, the cumulative experience over 1926-2005,
  2. the inflation round-trip, 1960-2005, from low- to high-inflation and back,
  3. rising inflation over 1960-1980, the “dark age” for bonds,
  4. disinflation, over 1981-2005,
  5. the last ten years, 1996-2005, the “golden age” for bonds, and
  6. the golden age for stocks, 1950-1965.

The range of episodes 2) to 5) clearly runs the gamut for bonds’ experience. We should also include an episode wherein stocks enjoyed especially favorable relative performance. Neither 1980-2005 nor 1996-2005 qualify as such, because bonds and equities both performed favorably. The best episode for stocks on a relative-performance basis is that of 1950-65, when stocks enjoyed a post-WWII bull market, while bonds were selling off in a prelude to the rising-inflation era.

Exhibit 3 shows compound average returns and risks (of total returns) for cash equivalents, intermediate T-Notes, long-duration corporate bonds, and large-cap stocks in each of these episodes. For each asset class in each episode, the ratio of average return to standard deviation is analogous to the Sharpe Ratio for each, and these ratios are instructive as to the efficacy of each asset. While stocks feature a higher average return than other asset classes, their returns are generally lower on a risk-adjusted basis. This is extremely important for constructing optimal portfolios in either framework.

Exhibit 4 displays the “optimal overlays” for each of the episodes listed above: that overlay in each episode with maximum Sharpe Ratio. Examining these across the various episodes provides a flavor of how optimal investing varies across market regimes. For example, optimally adding a percentage point of target return during the rising-inflation era (1960-80) would have involved a lot more investment in T-Notes and a lot less in long bonds than would have been the case during the falling-inflation era (1981-2005).

Exhibit 3
Expected Return and Total-Return Risk Over Various Historical Episodes
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset
Exhibit 4
Optimal Overlay Portfolio (z*) Over Various Episodes*
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset

Notice that the share of equities in these optimal overlays does not vary much across the different episodes. It is not even especially high during the “golden age of stocks” episode (1950-1965). In that episode, while the Sharpe Ratio for stocks exceeded those for bonds and notes, there was enough covariance between bonds and notes that risk would have been more efficiently taken in a long-short fixed-income position than in a leveraged equities position, and this is true even during stocks’ “golden age.”

Assuming that a liability valuation can be approximately fully hedged via a 100% bond allocation, we can identify the replicate portfolio on the total-return frontier (REP) in each episode by identifying the total-return-optimal portfolio with the same return as that of long bonds. Once again, subtracting these replicates from a 100% bond portfolio then identifies the impact of LDI on optimal portfolio composition—the amount of “shift” between total-return- and surplus-optimal portfolios—in each respective episode. Exhibit 5 displays these “shift portfolios.”

Exhibit 5
Shift Portfolios: Change in Portfolio Composition Due To Shift to LDI Framework
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset

As one might expect, an LDI framework would have had a large impact during the falling-inflation era. As shown in Exhibit 5, a total-return framework in that episode would have over-allocated to notes by 98% and to stocks by 36% and under-allocated to long bonds by 107%. The higher allocations to long bonds and cash and lower allocations to notes and stocks in a surplus-optimal framework would have produced the same total return, but with much lower surplus variability.4

It is surprising that the results for the two longer episodes, the 1926-2005 long-run and 1960-2005 inflation roundtrip, are similar. Both show very large impacts on optimal portfolio construction of moving to a liability mandate. In both of these longer episodes, LDI techniques would have featured dramatically higher allocation to long bonds than would a total-return-driven allocation, without sacrificing anything in total return. This is true despite the fact that long bonds were not stellar performers over those periods.

Actually, those results hold precisely because long bonds were not a stellar performer in those episodes. Because long bonds did not perform particularly well then, total-return-optimized allocations would have replicated bonds’ levels of returns via portfolios that were actually short long bonds—by quite a lot in the 1960-2005 episode (and total-return-optimal portfolios with other target returns would have been similarly under-allocated to bonds). The short positions in bonds in such “replicate” portfolios is an indication of how much a total-return mandate would have ignored the hedging benefits of long bonds in a surplus setting, and so it is an indication of how important a shift to an LDI mandate would have been.

Similarly, stocks’ golden age, 1950-1965, is also an example of an episode where a total-return perspective would not haven given consideration to long bonds. Thus, it is also an episode where the adoption of an LDI framework would have made a huge difference in portfolio allocation.

The two episodes where LDI would not have made much difference were bonds’ “golden age,” 1996-2005, and bonds’ “dark age,” the rising-inflation era of 1960-1980. During the “golden age,” bonds performed so well that the best way to replicate their return—even in a total-return framework—was via a bond-heavy allocation, so that adopting a liability benchmark, ironically, would not have had much impact. During bonds’ “dark age,” their performance was so poor that a total-return strategy choosing to replicate their return would have had to efficiently do so via a bond-heavy allocation, so that a shift to a liability benchmark would not have had much impact there either.

Notice also that the shift portfolios feature a substantial reduction in equity exposure in all episodes except those of rising inflation and stocks’ golden age. The shift portfolio for the stocks’ golden age episode features both a large shift into long bonds and essentially no change in equity allocations.

Exhibit 6 applies these results by showing the LDI counterpart in each episode to a standard 10/30/0/60 portfolio (60% equities, with fixed-income exposure kept in short maturities). For each episode, we took the “shift portfolio” from Exhibit 4 and added it to the “standard” 10/30/0/60 portfolio.5

Exhibit 6
LDI Alternatives to 60% Stocks/40% F.I. Standard Portfolio
the-impact-of-ldi-on-portfolio-selection-2006-05
Source: Ibbotson Associates, Western Asset

A common objection to the allocations in Exhibit 6 is that they involve leverage. However, a pension fund necessarily involves leverage. It is 100% short its liabilities and long its asset allocation. A standard, 10/30/0/60 allocation is, in truth, 10/30/-100/60, borrowing long-term bonds in order to invest in stocks and short-duration paper. The gross value of long positions may be higher in the “leveraged allocations” in Exhibit 6 than in a standard pension allocation, but these are in less-risky asset categories, and the aggregate surplus risks of those positions are sharply lower than those in the equity-heavy allocations traditionally thought of as “non-leveraged.”

IV. Pension Management and Portable Alpha Management

Our characterization of surplus-optimal allocations as combinations of the liability-hedge and exposure in an optimal overlay makes pension optimization identical to managing a portable alpha position. The “beta” position here is merely the liability-hedge. Managing a portable alpha strategy in, say, an S&P500 beta involves replicating the equities exposure (the beta) via derivatives (S&P futures or swaps) and porting in the alpha source with maximum incremental return relative to tracking error (incremental risk). In the same way, if one were to choose a portable alpha strategy with a beta position equal to the liabilities of a particular pension plan, the alpha source in that strategy would serve exactly the same role as would the overlay added onto the liability-hedge in a surplus-optimal allocation.6

A formal portable alpha position typically involves active management relative to an appropriate benchmark. Thus, in Western Asset (2006a), we described the merits of porting active short-duration fixed-income onto large-cap equity beta positions. In that case, an “information ratio” was relevant, measuring the return and risk of a manager’s performance relative to this benchmark.

If a particular manager could be expected to produce especially favorable information ratios, there is no reason why allocations to his fund couldn’t be taken as the “optimal overlay” to “port onto” a liability-hedge in order to construct optimal pension allocations. In general, for optimizing pension allocations, seeking “sector bets” with favorable Sharpe Ratios or funds with favorable information ratios has the same effect. Pursuing maximum Sharpe or information ratios in pension allocation is interchangeable.7

V. Conclusions

Utilizing the decomposition property of optimal portfolios allowed us to identify a single “shift portfolio” in each historical episode that fully described the impact of shifting to an LDI framework within that episode, regardless of how aggressive a plan’s target return was. We found this “impact of LDI” to be generally stable across historical episodes. Under extreme conditions, the prominence of long bonds in surplus-optimal allocations might decline, or the prominence of equity allocations might rise to their standard 60% level. In most episodes, however, surplus-optimal allocations featured long bond allocations sharply higher and equity allocations substantially lower than what total-return (traditional) optimal portfolios contained.8

Our empirical results were derived for an asset universe containing cash equivalents, intermediate TBonds, long-term corporates, and large-cap equities. However, the decomposition results we utilize will hold for any “universe” of assets. Introducing new assets will change the results only to the extent that the new assets change the profile of the optimal overlay (z*) and to the extent that they alter the liability-hedge. And if the “added” assets do not affect the liability-hedge (that is, if they are not correlated with liabilities, once the correlation of long bonds with liabilities is allowed for), then they likely will not alter the “impact of LDI” (the shift portfolio) as we have defined it.9

In any case, it would be interesting to see how the addition of commodities or real estate into the asset mix might alter the results stated here. Also, our decomposition result lends itself to a simple exposition of the impact of different funding levels on optimal pension allocation. We’ll save these topics for future reports.

Appendix. A Verbal Derivation of the Mutual Fund Theorem for Surplus Optimization

In this Appendix, we elaborate on the Mutual Fund Theorem of traditional portfolio theory, we extend it to the LDI or surplus-optimization framework, and then we derive the result stated in the text that surplus- and total-return-optimal portfolios differ by the same “shift” portfolio regardless of the target level of returns. In order to expedite this analysis, we’ll introduce a simple analytical shortcut.

We discuss here various cases of moving from one portfolio to another, either optimizing across a range of portfolios or moving from one optimal portfolio to another along an efficient frontier. Such movements between portfolios can equivalently be described as the addition of an overlay portfolio to the initial portfolio. If any two portfolios w1 and w2 both have net asset allocations equal to 100% of assets, then w2-w1 must net to 0% of assets, and so the movement from w1 to w2 can be described as the addition of the overlay portfolio w2-w1 to w1. (An overlay can be thought of as a swap or a combination of swaps.)

Now, the Mutual Fund Theorem—or separation theorem—of Markowitz, Sharpe-Lintner, Merton, et al. states that all portfolios on the traditional, total-return, efficient frontier can be decomposed into the same two base portfolios—or “mutual funds.”10 Merton’s (1972) statement of this theorem represented the two “base” portfolios as that portfolio on the efficient frontier with zero expected return and the same (maximum Sharpe Ratio) overlay that we have described in the text. By adding sufficient exposure in the optimal overlay to Merton’s zero-return portfolio, one can arrive at the minimum-variance portfolio (MVP), or rightmost point on the total-return efficient frontier. One can then describe each portfolio on that frontier as decomposing into this MVP plus the optimal overlay z*, and this is the representation of the frontier that Keel and Muller (1995) utilize. They also extend this result to surplus-optimization and show that surplus-optimal portfolios can be expressed as composites of the minimum-surplus-variance-portfolio and an overlay.

All we have done is expand their results to 1) explicitly show that the overlays for the total-return and surplus-optimal frontiers are 2) identical, identify z* as the overlay with maximum Sharpe Ratio, and then 3) exploit these findings to identify the impact of LDI in various settings. We can verbally demonstrate the decomposition property relatively simply. Let’s take the surplus-optimization case first.

We want to show that any surplus-optimal allocation can be expressed as a combination of the liabilitiy hedge (MSVP) and the overlay with maximum Sharpe Ratio. First off, notice that the portfolios on the efficient frontier have maximum ratio of expected return to surplus risk (standard deviation). We’ll describe this as maximum surplus-Sharpe Ratio. That is, maximizing expected return for a given level of surplus risk is the same thing as maximizing the surplus-Sharpe Ratio.

Now, the MSVP is clearly on the efficient frontier, since it has the lowest surplus-risk of ANY available portfolio. Moving from the MSVP to any other point on the efficient frontier merely involves adding an overlay. It stands to reason that in order to maximize the surplus-Sharpe Ratio for the resulting portfolio, we should add that overlay portfolio which itself has maximum Sharpe Ratio.

A technical consideration is that in adding an overlay, we change the surplus-Sharpe Ratio of the result (we change the surplus-risk of the resulting portfolio) both through the Sharpe Ratio of the overlay and also through the covariance of the overlay with the surplus for the MSVP (call this the “surplus covariance” for the MSVP). Well, the surplus-covariance of the MSVP with any overlay must be zero, by virtue of the fact that the MSVP has minimum surplus-variance.

(The surplus covariance of any portfolio with any overlay is identical to the change in surplus variance due to a portfolio shift in the direction of that overlay. Since the MSVP is a minimum, it must have zero rate of change in surplus variance for very small portfolio shifts in any direction. Its derivative must be zero in any direction. So it must have zero surplus covariance with portfolio shifts in any direction. Therefore, it must have zero surplus covariance with any overlay.)

Since the MSVP has zero surplus-covariance with ALL overlays, adding on that overlay with maximum traditional Sharpe Ratio will necessarily maximize the surplus-Sharpe Ratio of the result.11 Therefore, starting with the minimum surplus-variance portfolio and adding various levels of a maximum Sharpe-ratio overlay will necessarily reproduce optimal portfolios along the efficient frontier for surplus optimization.

For total-return-optimal portfolios, the portfolio with minimum asset variance also has zero (normal) covariance with any overlay (by virtue of its being a minimum). Therefore, total-return optimal portfolios can be expressed as combinations of the minimum-variance portfolio and maximum-Sharpe-ratio overlay.

From these results, total-return-optimal portfolios can be expressed as MVP+k·z*, for some k, and surplus-optimal portfolios as MSVP+m·z*, for some m. For the replicate total-return portfolio with the same expected return as the MSVP, REP = MVP+k1·z*, for some specific k1. It then follows that the difference in portfolio weights between any two total-return- and surplus-optimal portfolios with the same expected return is merely the fixed overlay MSVP–MVP– k1·z*. (Remember that we have defined z* such that it has expected return of 1 percentage point. Since z* is an overlay, various levels of exposure in it will also be overlays, and so we can define any one “intensity” of it as being the overlay z*.)

For a formal derivation of these results, Email us at:

Michael.Bazdarich@westernasset.com

.

References

Keel, Alex and Heinz H. Muller 1995, “Efficient Portfolios In the Asset-Liability Context,” Astin Bulletin, November

Merton, Robert 1972, “An Analytic Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis (also available on Merton’s personal website).

Sharpe, William F. and Lawrence G. Tint 1990, “Liabilities—A New Approach,” Journal of Portfolio Management, Winter.

Western Asset Management (2006a), “Which Alpha Would You Choose?.”

Western Asset Management (2006b), “Active Liability Driven Investing and Pension Management.”

  1. Sharpe and Tint (1990) use this language. As we discussed in Western Asset (2006b), the perception is still common in practice here and abroad that LDI tactics abandon attempts to obtain favorable portfolio returns, a perception we characterize as “misperception.”
  2. See Western Asset (2006a) for an explanation of portable alpha techniques.
  3. The two efficient frontiers are based the 1960-2005 returns experience of an asset universe of cash-equivalents (1-month TBills), intermediate TNotes, long-term corporate bonds, and large-cap stocks (the S&P 500), with returns on all these assets as reported by Ibbotson and Associates. This is exactly the same asset universe that was analyzed in Western Asset (2006b). As in the preceding report, the surplus-optimal efficient frontier is also based on a liability valuation that varies with long-term corporate bonds and with a non-systematic variation of 2% per year. (The previous report assumed a non-systematic variation in liabilities of 4% per year.) This same asset universe and liabilities valuation are used across the set of historical episodes analyzed in the rest of this report.
  4. Adding these shift portfolios to total-return optimal portfolios results in a fixed reduction in surplus variance. However, the standard deviation of surplus does not decline by a fixed amount for all optimal portfolios [since (x2+y2)1/2x depends on the value of x even if y is constant]. So the reductions in surplus risk reflected in the bottom lines of Exhibit 4 are only suggestive of the reductions for other levels of target return.

  5. The “standard” 60% equity portfolio is a handy reference point for this comparison. However, this standard portfolio would not have been optimal even under a total-return framework in any of these episodes. Adding the shift factor between total-return- and surplus-optimal portfolios to a non-optimal (total-return) portfolio yields a portfolio that is not surplus-optimal, and so the portfolios listed in Exhibit 5 are not surplus-optimal. They only suggest what a surplus-optimal analogue to the “standard” portfolio would be.
  6. In considering the combination of a portfolio and an overlay, it is arbitrary which we consider the overlay and which the portfolio. Thus, a portable alpha position with a beta in the S&P500 can be constructed via S&P futures (an overlay) and a long position in the alpha source or via a long position in the cash S&P and a derivative position (overlay) in the alpha source. In the case of surplus optimization and, thus, of a liabilities “beta,” that beta hedge can be achieved either by an actual (dedicated-bond) portfolio or via a swap (overlay), and the optimal overlay could be pursued either in true overlay form or with a long cash position added back to it. It makes little difference which side of the allocation is the actual “overlay” and which is the “full portfolio.”
  7. If a variety of “alpha sources” are chosen for the active pension allocation, be they sectors or individual funds, one should take account of the correlation across these sources, as well as their respective information ratios, so as to insure these sources are “optimally” combined.
  8. Our results were derived for one-year holding periods. Interestingly enough, over 5- and 10-year holding periods, using the 1926-2005 experience, the relevant “shift portfolios” comparable to those in Exhibit 4 show even higher long-bond weights than reported for the one-year holding periods. Going from 1- to 5-year holding-periods, the optimal share for equities drops 4.5 percentage points, and moving to 10-year holding periods reduces it a further 5.8 percentage points. This reflects that fact that because bonds perform more poorly over the longer holding-periods on a total-return basis, the impact of moving to an LDI framework is all the greater. Still, the one-year holding-period shown in the text is relevant for private-sector DB plans under FASB regulations.
  9. If adding assets to the asset universe does not affect the full hedge of liabilities, say if it remains a 100% bond allocation as assumed in the text, then the new assets will alter the shift portfolio only to the extent that they affect the “replicate” portfolio.
  10. All the popular-consumption financial websites we have come across define the Mutual Fund Theorem as stating that investors should “choose to invest their entire risky portfolio in a market-index or mutual fund.” While this result does follow from our statement of the theorem, our statement is what can be found in actual finance-theory treatises.
  11. The surplus variance of the MSVP plus an overlay will be the surplus variance of the MSVP plus the “normal” variance of the overlay plus the surplus covariance of the MSVP with the overlay, this last term being zero. Since the overlay adds to surplus variance according to its “normal” variance, it is the “normal” Sharpe Ratio for the overlay that matters in affecting the surplus-Sharpe Ratio of the resulting portfolio.
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