Duration Positioning in Anticipation of a Parallel Upward Shift in the Yield Curve

Suppose rates are going to fall by 50 bps. You just know it. Your fixed-income portfolio is about to grow, and that's great. You know that altering your portfolio's duration would be a good move today, so it's time to try it.

To start with, which direction should you most likely change duration?

That's probably the wrong way. More duration means more power to the yield curve change, and a lower yield curve isn't something you want to shy away from. This is an increase in market value that you want to take advantage of. Get that duration moving up.

Of course. Get that duration moving up.

Here's your portfolio: | Maturity | Coupon | Price | YTM | Market Value | Effective Duration | |-----|-----|-----|-----|----|-----| | 3 | 3.2 | 100 | 3.20% | 250,000 | 2.82 | | 4 | 3.7 | 100 | 3.70% | 250,000 | 3.66 | | 5 | 3.9 | 100 | 3.90% | 500,000 | 4.46 | | Portfolio | | | 3.68% | 1,000,000 | 3.85 | Now this 50 bp decline is expected to take place sometime in the next year. So that's your holding period return of focus right now. Knowing that the two-year rate is 2.70%, what changes do you expect to see in your bond prices over the next year, without that yield curve shift?

No. There will be a change. Consider the "roll down."

Yes! There will be price increases as you roll down the yield curve. These simple, hypothetical bonds are all priced at par because they all pay coupons that match their yields. Next year, assuming a fixed yield curve, that won't be the case. The coupons will look "too good," and you'll enjoy some price appreciation while you experience rolldown.

No. There's no reason to expect price declines. Look at the implied yield curve now.

Here's what that would look like, adding now the prices after rolldown and the holding period returns: | Maturity | Coupon | Price | Price after Rolldown | HPR | |-----|-----|-----|-----|-----| | 3 | 3.2 | 100 | 100.96 | 4.16% | | 4 | 3.7 | 100 | 101.41 | 5.11% | | 5 | 3.9 | 100 | 100.73 | 4.63% | The focus here isn't on these calculations, but do you want to see some specifics about how these values were obtained?

Sure. The price after rolldown is found by moving forward one period in your mind. Of course the four-year bond is at par since it's paying 3.7% coupons and yielding 3.7%. But go forward one year with the same yield curve, and now the YTM is 3.2% for the bond, which now has three years to maturity. So the price is 101.41. $$\displaystyle P = \frac{3.7}{1 + 0.032} + \frac{3.7}{(1 + 0.032)^2} + \frac{100 + 3.7}{(1 + 0.032)^3} = 101.41 $$ Same idea for the other bonds. Just go up one row for the yield for discounting (remember that the two-year is expected to be 2.2% to continue the simple linear yield curve), and you'll find those prices. Then, the holding period yields are found by just using the fact that you'll get a coupon plus this new price as the return for starting with the beginning price of 100. Again, to show this with the four-year bond, $$\displaystyle HPR = \frac{3.7 + 101.41}{100} - 1 = 0.0511 = 5.11 \% $$.

Okay.

With this in mind, now consider the decision to shift weights in anticipation of the yield curve shift. How would you want to evaluate your choices about how to reweight the portfolio?

No. That would only be best if you knew the shift would happen suddenly, and within a few hours. You really can't assume that.

Exactly. So use the same idea here. Move forward a year, now assuming the yield curve does shift downward as expected, and you can see the estimated HPR for each bond: | Maturity | Coupon | Price | Price After Rolldown and -50 bp Shift | HPR | |-----|-----|-----|-----|-----| | 3 | 3.2 | 100 | 101.94 | 5.14% | | 4 | 3.7 | 100 | 102.85 | 6.55% | | 5 | 3.9 | 100 | 102.59 | 6.49% |

No. That would miss the main point here that the timing of the shift is unknown, and there may be other things to consider.

Surprise! The best choice isn't the five-year bond as you may have assumed from the start, even though it had the highest duration. The holding period return was lower without the yield curve shift, and it's lower _with_ the yield curve shift—just a little. What do you think is a reason why this would be the case?

No. That really doesn't matter as far as the reason behind this HPR difference.

Not really. That doesn't seem to make the difference. With or without the decline, the HPR is lower for the five-year bond than for the four-year bond.

Yes! Remember one detail of rolling down the yield curve: it works best when the curve is steep. That four-year bond will roll nicely, while the five-year bond won't roll as much. So there's not as much benefit from having such a high weight in that bond. That's the main force behind the lower HPR for that issue.

So now the final step: adjust. Go back to your current weighting and the relevant numbers from your expectations. Perhaps you have a mandate that duration needs to stay between 3.80 and 4.00. You can look back to see that you're currently at 3.85. As you optimize, which one of these will be a binding constraint?

Good call!

Actually, no. It's the minimum duration of 3.80.

This may seem odd: Didn't this whole thing start with "the yield curve is going down, so get that duration moving up"? Yes, it did. And it's more likely that you will want duration to increase when you expect a yield curve decline. This example was chosen to highlight the importance of accurate forecasting, especially when you're constrained to a certain investment horizon. But now you have discovered that the best bond to hold over the next year, either with the curve drop or not, is that four-year bond. So you want to pile as much money into that bond as you can. That bond's duration is 3.66, so your portfolio duration will go down.

There's no need to look at the three-year bond at all. The duration is even lower, and it offers a lower HPR. So forget it. Instead, just use the other two bonds to meet your investment constraint. Out of the 1,000,000 market value, how much will you invest in the five-year bond?

No, that's too much. You'd end up with a duration of more than 4 and not optimize your expected returns.

No. A 50% investment in each of those two bonds would leave you with a duration that's outside of the required range.

That's right. Just use a simple average weighting to get to your duration of 3.8. $$\displaystyle w(3.66) + (1 - w)(4.46) = 3.80 $$ With a little algebra, this turns out to leave you with a four-year bond weighting of 82.5% and a five-year bond weighting of 17.5%. That's a significant shift from the current allocation. But as you can see, with or without the shift, you'll end up with a greater return in a year.

To summarize: [[summary]]

Up

Down

None

Increase

Decrease

Yes

No, it's fine

Just use the five-year bond since it has the highest duration

Look at the estimated HPY for each bond using the rolldown and yield curve shift

Estimate the effect of the yield curve shift on current prices, without looking forward

The yield curve flattens there

The yield curve is expected to decline

That bond has a duration above that of the portfolio

At least 3.80

No more than 4.00

Close to 20%

Close to 50%

Close to 80%

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