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Suppose that you are the strategist for an aggressively (and actively) managed “long-only” portfolio of Treasury notes and bonds, having maturities spread out along the yield curve. You have to stay fully invested in Treasuries, cannot use derivatives, cannot take on short positions, and cannot use borrowed funds for leverage. You are authorized to buy and sell securities opportunistically in light of your view on an impending change to the level and shape of the Treasury yield curve. Your objective is to maximize quarterly returns, that is, achieve gains (when Treasury yields go down) as well as to minimize losses (when yields go up). Currently, the market-value-weighted average modified duration of your portfolio is 7.10 and the average convexity is 110.5 (using the Bloomberg convexities times 100). These are descriptive statistics and are averages of the yield durations and convexities.

Some decisions are quite straightforward. Suppose your view is that the Treasury yield curve will shift either upward or downward in a parallel manner, as illustrated in Figure 10.1. Surely when you anticipate the upward shift, you sell some long-term T-bonds and buy short-term T-notes, thereby reducing average duration. When your view is a downward parallel shift, you sell short-term T-notes and buy long-term T-bonds in order to increase the overall modified duration of the portfolio. That sounds simple enough, but in deciding how much to buy and sell, you have to consider trading costs and the chance that your rate view turns out to be incorrect. True believers in the expectations theory of the term structure will be quick to remind you that market expectations about factors driving yields up or down are already priced into the bonds.

The key point is that average duration is a perfectly acceptable summary statistic for the aggressive strategy. That is, you could instruct your traders to bring the average duration down to 6.10 from 7.10 in the first scenario or to take it up to 8.10 in the second. The change in the duration statistic captures the extent of the aggressive restructuring. For this purpose,

Parallel Upward and Downward Yield Curve Shifts

FIGURE 10.1 Parallel Upward and Downward Yield Curve Shifts

the standard market-value-weighted average of the yield duration statistics is fine. You are not estimating the change in market value – for that you would be better off using the weighted average of curve durations. Notice that your trades also change average convexity, but that is not the focus of your strategic move.

Although the assumption of a parallel shift to the yield curve is common (and might even allow for arbitrage opportunities), it is an abstraction from reality. In fact, most yield curve events entail upward or downward shifts combined with some steepening or flattening. Figure 10.2 illustrates these four shifts – bull and bear steepeners in the upper diagram and bull and bear “flatteners” in the lower diagram. A bull market means yields go down in general and bond prices rise; a bear market is when yields rise and prices fall.

Steepening and Flattening Yield Curve Shifts

FIGURE 10.2 Steepening and Flattening Yield Curve Shifts

Our problem is that in two of the four scenarios, the change in average duration no longer serves as a reasonable descriptive statistic for aggressive strategy. In those circumstances, you cannot instruct your traders to execute buy-and-sell orders to change overall average duration by a certain amount. In fact, it's not always obvious what you should sell and what you should buy. Do you see the two scenarios when average duration “fails” as a summary statistic for strategy and the two in which it continues to work well, as in the case of a parallel shift? As you think about this, remember that the estimated change in the market value of a bond (or a cluster of bonds at a particular point on the yield curve) is the modified duration times the market value (that product being the money, or dollar, duration) times the change in the yield.

Look first at the bear steepener in Figure 10.2. Yields on long-term T-bonds are expected to rise more than yields on short-term T-notes. You know what to do: Sell the former and buy the latter, thereby reducing average duration. Our problem is the bull steepener because short-term T-note yields are expected to fall more than the yields on long-term T-bonds. Market values at each point on the yield curve go up – it's a rising-price bull market. However, the extent of the increase in market value depends critically on the extent of the change in yield. The short end of the curve has larger reductions in yields but lower durations; the long end has smaller changes in yields but higher durations. This is not to say that as a strategist you are stymied, just that you need to factor into your analysis the extent of the anticipated change in yield at the various points on the curve. Note that partial (or key rate) durations would come in handy here. In principle, you could end up increasing or decreasing average duration.

Now consider the bull flattener. It's clear that you want to sell some of your short-term positions to load up on the long end. Not only are the long-term T-bonds expected to experience the larger drop in their yields to maturity, they also have much higher modified durations. In this scenario you could tell your traders to increase average duration as well as convexity. The problem is with the bear flattener. Which positions will lose more value – the short-term T-notes that have the larger increases in yield but the lower durations, or the long-term T-bonds that have the smaller changes in yield but higher durations? Again, you need a more articulated rate view to provide instructions for your traders. Changing average duration by a certain amount is not enough.

Which of the four nonparallel shifts are most likely to occur? Recall from Chapter 5 the three stylized facts regarding the term structure of interest rates – the normal upward slope, usually (but not recently) greater rate volatility at the short-term end of curve, and positively correlated shifts up and down. When the curve is steep due to low short-term rates, the tendency is for rising rates and a flatter curve. When the curve is inverted because of high short-term rates, “regression to the mean” leads to lower rates and a steeper (more upwardly sloped) curve. These imply that the two most likely shifts are bull steepening and bear flattening – the two for which average duration does not provide a clear and simple guide to portfolio strategy. I realize that this is disappointing news to those of us who are inclined to think that Frederick Macaulay was a prophet and duration is imbued with mystical qualities.

Figure 10.3 shows two examples of changes in the shape of the yield curve. Sometimes we actually do see U-shaped curves in which medium- term maturities have the lowest yields and humped curves where they are highest. Consider first the “negative butterfly” twist to the yield curve portrayed in the upper diagram. Short-term and long-term yields are expected to fall while medium-term yields rise. Given this rate view, you want to sell the medium-term T-notes and buy both short-term T-notes and long-term T-bonds.

Shape-Changing Yield Curve Shifts

FIGURE 10.3 Shape-Changing Yield Curve Shifts

The key point is that once again you cannot use the change in average duration to describe the strategic rebalancing. Your negative butterfly trades could be duration neutral. However, transactions to “sell the belly and buy the wings” increase average convexity. You have increased cash flow dispersion by putting more weight at the ends of the yield curve. In this case, you could instruct your traders to keep average duration around 7.10 but raise average convexity from 110.5 to about 120. That signals how much restructuring you want for the portfolio.

I'm sure you now can deduce what to do with an anticipated “positive butterfly” shift in the lower diagram in Figure 10.3. You sell the wings, reducing your holdings of both short-term and long-term Treasuries, and add to the medium-term positions that you anticipate appreciating in value. Once again, the change in average duration is not obvious; however, you will be reducing average convexity and cash flow dispersion. You could charge your trading team to stay duration neutral but lower average convexity from 110.5 to about 100.

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