Yield Curve Dynamics
Yield curves change. Quite a bit.
If you've ever seen a fast time-lapse video of a yield curve, it can look like a whip moving around erratically. To try to tame this dynamic representation of yields in varying time periods, three movement types are generally used for classification: parallel shifts, steepness twists, and curvature changes. Following the 2008 financial crisis, which direction did most parallel yield curve shifts go?
No, most went down.
Absolutely.
Way down. In fact, once zero was reached, everyone got excited about the possibility of negative interest rates. It got a lot less exciting month after month when some rates stayed negative. Here's a look at the 10-year Treasury Yield from just prior to the financial crisis to 2022, also showing a similar drop during the Covid flight to quality:
The 10-year Treasury rate didn't go into negative territory, but the 5-year Euro area government bond rate did. It's far from normal, but pretend for a moment that rates stayed negative. A coupon bond could actually have a negative coupon rate. Beyond the typical credit risk from the issuer, there would be credit risk from the bondholder who is responsible for making periodic payments for the privilege to essentially lend the face value. Strange. Then, there's duration. What would negative coupon rates do to the bond's duration?
No, it couldn't be equal.
Consider both a typical positive coupon bond and a zero-coupon bond.
Yes.
The coupons would be less than zero, so it would have a duration greater than that of a zero-coupon bond. Again, strange. But if negative interest rates are part of the common future landscape of finance, such things could happen.
No, it couldn't be less.
Think about the duration of a zero-coupon bond, as well as a standard positive coupon bond.
Steepness is often measured in basis points, as the yield of a 30-year government bond minus the yield of a 2-year government bond. A "normal" yield curve might have a 2% 2-year rate and a 5% 30-year rate. Then the spread is 300 basis points. If the short-term rate suddenly rises while the long-term rate doesn't, the curve will flatten; if it continues until it's above 5%, then the yield curve is inverted.
The short-term rate is usually much more volatile than the long-term rate. How might you relate the level of the short-term rate with the steepness of the curve?
Exactly!
If the 30-year rate were very stable, then the short-term rate would determine the steepness. Start with a normal curve. As short-term rates increase, the curve flattens and even inverts. As the short-term rate falls, the curve becomes steeper.
This steepness measure for US Treasuries has changed quite a bit over the past 40 years, with this differential being negative once per decade on average. Here is the period since just before the financial crisis, the last time that the yield curve inverted in this way:
Not quite.
This would actually suggest the opposite for relative volatilities.
No.
They would be clearly related if this relationship held to an extreme. Consider an unmoving 30-year rate, for example.
Finally, there's curvature. Steepness gives you the "long and the short" of it, so to speak, but the yield curve can be straight, curved, flat, or humped, which depends on that 10-year yield. If it changes independently, the yield curve is like a butterfly flapping its wings. The __butterfly spread__ is a measure of just how much the medium-term yield does or doesn't "fall in line" with the others.
Suppose that the yield curve was linear, no hump or any disturbance in the middle. What would the medium-term yield be, relative to the short-term and long-term yields?
Probably not.
Assuming positive yields everywhere, that would lead to a hump.
No.
That wouldn't be possible; those two rates are probably different from each other.
Precisely.
Just add up the long-term and short-term yields and divide by 2. If that were the case, then the butterfly spread would be zero. So with a little algebra, this same idea can be placed into a simple equation.
$$\displaystyle \mbox{Butterfly spread} = -(\mbox{Short-term yield}) + (2 \times \mbox{Medium-term yield}) - \mbox{Long-term yield} $$
Just for clarity, what would a higher short-term rate do to this measure, all else being equal?
No, actually it would reduce it.
You got it!
With other rates unchanged, this higher short-term rate would cause the curve to "sag" more. This same idea is caught in the butterfly spread, since the short-term and long-term rates both enter negatively into the measure. Negative butterfly spreads show this sag, while a positive spread would show a more common "hump" to the curve. Here's how common each have been over the past 35 years, again using US Treasuries:
Yield curves really do change. Quite a bit.
To summarize:
[[summary]]
Up
Down
Make it equal to maturity
Make it greater than maturity
Make it just a little less than maturity
The two should be unrelated
Lower short-term rates mean a flatter curve
Lower short-term rates mean a steeper curve
The sum
The same
The average
Reduce it
Increase it
Continue
Continue
Continue
Continue
