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Immunization theory is based on the big-bundle-of-cash-flow portfolio statistics. In the example, we lock in the cash flow yield of 3.364% over the 12-year horizon because the portfolio Macaulay duration is 12.030. But that duration statistic is not commonly used in practice; instead it typically is estimated using the market-value-weighted average of the individual yield durations. In Chapter 9, we calculated AvgMacDur(Yield) to be 11.224, considerably below the “true” portfolio duration because of the steep upward slope to the yield curve.

Bond strategies in practice inevitably have some degree of model risk arising from the statistics that we use to structure and manage the portfolio. Typically, we estimate those statistics using market-value-weighted averages. Sometimes they are accurate (e.g., when the yield curve is flat), but most of the time those averages introduce measurement error. For an immunization strategy, one way of mitigating the model risk as well as interest rate risk arising from not-so-well-behaved yield curves (i.e., nonparallel shifts, especially bear steepeners) is to build the portfolio to resemble that which works best – a zero-coupon bond maturing at or close to the horizon.

The standard textbook prescription to mitigate risk in an immunization strategy is to match the duration of a portfolio of coupon bonds to the horizon and to minimize cash flow dispersion. The intuition here is obvious – reducing dispersion by clustering the maturities of the bonds near the horizon makes the portfolio look more like the zero. In the extreme, DispPORT, the portfolio dispersion statistic introduced in Chapter 9, approaches zero. This not only reduces both cash flow reinvestment and market price risk, it also reduces model risk. In a tightly clustered, low-dispersion portfolio, the market-value-weighted averages are closer to the “true” portfolio statistics, even in a steep yield curve environment. For instance, AvgMacDur( Yield) is closer to MacDurPORT, the lower is cash flow dispersion.

The problem with the prescription to duration match and minimize dispersion is not in theory; it is in implementation. In Chapter 9,1 note that dispersion is not a statistic reported on Bloomberg (although it could be) and, more important, even if it were readily accessible, the market-value-weighted average of the statistics on individual bonds can be misleading. Fortunately, equation 9.7, repeated here as equation 10.4, offers a handy remedy.


For a given Macaulay duration for the portfolio, minimizing portfolio dispersion is equivalent to minimizing portfolio convexity. Note that this result is for the “true” statistics, not the weighted averages. However, given that it is common to calculate AvgConvexity{Yield), the market-value-weighted average of the individual yield convexities, my suggestion is that an immunization strategy minimizes interest rate and model risk by duration matching

and minimizing average convexity. On the other hand, a passive-aggressive approach to immunization is to match the duration to the investment horizon and, instead, maximize convexity. That will outperform the “extreme- zero-replication” approach to the extent that the yield curve is well behaved and does not twist in undesired ways.

There actually are practical applications for an immunization strategy to target a nominal rate of return. An example is defeasance whereby an issuer, often a corporation or municipality, has the cash to pay off debt liabilities and improve its leverage ratio and perhaps get an upgrade to its credit rating. However, unless the debt currently is callable, it can be difficult and expensive to buy back the bonds via a tender offer or an open market repurchase program. Seasoned corporate and municipal bonds are not very liquid and reside in hold-to-maturity portfolios. However, if the cash is invested in high-quality bonds, for instance, Treasury and agency notes, there are circumstances when for accounting purposes both the debt and the bond investments can be taken off the balance sheet. One way of building the defeasance portfolio is to match coupon and principal cash flows as closely as possible. Another is to match the durations and then actively rebalance to stay on duration target.

Many real-world investment problems are better described as attempts to target a real, i.e., after inflation, rate of return, for instance, on retirement savings. The same principle of immunization applies, only now it is to match the real rate duration to the horizon. Suppose that an individual plans to retire in 20 years. A 20-year P-Linker, such as TIPS, matches the horizon but still leaves coupon reinvestment risk on the table. Granted, the volatility in real rates is less than in nominal rates, but still a lower-risk strategy is to build a portfolio of TIPS that has an average real rate duration of 20 and then as time passes to stay on real rate duration target.

Recall from Chapter 7 that an advantage of P-Linkers is that the real rate duration statistic is high relative to traditional fixed-rate bonds and is not a function of the inflation rate. C-Linkers, however, are problematic for this strategy because their real rate durations are not only lower but, more importantly, depend on the projected inflation rate. It's much harder to build a real-rate immunizing portfolio using C-Linkers.

Another practical problem in implementing immunization strategy is horizon risk. This is the risk that once the portfolio is built, the investor's plans change, requiring early liquidation of the bonds or unexpected cash flow reinvestment. I think the biggest challenge that financial planners have is to get investors to reveal their financial objective, their true investment horizon, and their ability to stay committed to that time frame. Frankly, once the amount of desired total return and the time horizon are known, building a duration-matching bond portfolio is relatively easy.

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