A Framework for Evaluating Yield Curve Trades

Consider this five-piece decomposition of fixed-income returns. $$\displaystyle E(R) \approx \text{Coupon income} + \text{Rolldown return} $$ $$\displaystyle +/- E [\Delta \text{Price due to investor's view of benchmark yield}] $$ $$\displaystyle +/- E[\Delta \text{Price due to investor's view of yield spreads}] $$ $$\displaystyle +/- E[\Delta \text{Price due to investor's view of currency value changes}] $$. You've seen this before. The first two pieces are the coupon divided by price (coupon income), and then the return you get from buying something like a two-year security with a higher yield and holding it until it becomes a one-year security with a lower yield (rolldown return). These sum to become the rolling yield.

Which do you think is determined by the shape of the yield curve?

No. One of these directly benefits from an upward-sloping yield curve that stays still.

That's right. The rolldown return, staying with the one-year and two-year idea, requires that the two-year yield is higher than the one-year yield, and those yields remain unchanged for a year. If so, then buying a two-year bond for a year gives you a higher total return than either of those two yields, due to the price change.

No. A coupon bond will still have a coupon divided by price, and that price can be obtained with many different yield curves.

For example, suppose a two-year zero-coupon bond has a yield to maturity of 2.5%. The one-year yield is just 2.0%. If the yield curve stays where it is, and as time passes the two-year bond becomes a one-year bond, that means that the price of the two-year bond will change over the next year from $$\displaystyle P_0 = \frac{100}{(1.025)^2} = 95.1814 $$ to $$\displaystyle P_0 = \frac{100}{1.020} = 98.0392$$. That's a rolldown return of 3.0025% from the price appreciation. Now if you see yields of 2% and 2.5% and can make a 3% return, you might want to ask if there's a catch. There is. What can you say about a forward rate in this case?

No. You can calculate a particular forward rate in this case.

Exactly. The one-year forward rate one year from now is implied by these rates: $$\displaystyle f(1,1) = \frac{1.025^2}{1.02} - 1 = 0.0030025 = 3.0025 \%$$. So it's already there, in a way. The market already knows about this difference. So what does that imply about the one-year yield today?

No. That's an average, but consider how you would bootstrap the only forward rate possible here.

Yes.

No. Actually it should rise. The higher forward rates suggest that future spot rates should move in that direction: up.

Now it makes sense. You get the special 3% return if you see that the market expects rates to rise, you don't think the yield curve will change, and you're right. That's the key to getting this special return. Now for kind of a tricky question. Think about this: If a manager tries this and buys the two-year bond for a one-year holding period, and rates do rise, does that change the rolling yield?

Not really, no.

Good job!

If you look back at the decomposition, you'll see that the first two components that sum to make the rolling yield aren't in expectation: they are actual. How is it possible to come up with an expected return today if you need actual figures for the rolling yield? Well, it's because they are all assumed constant. The coupon and current price are known, so the coupon income is easy. The rolldown return is calculated under the assumption that the yield curve is unchanged. So that's how you get around expectations there. The rolldown return in this case is 3.0025%, no matter what happens. Any expected yield curve changes go in the third component.

This third component of expected yield curve changes is used with duration and convexity. $$\displaystyle E[\Delta P_{\text{Bond}}] = -MD \times \Delta \text{Yield} + \frac{1}{2} \times \text{Conv} \times (\Delta \text{Yield})^2 $$ You've seen that before. Nothing new. Although, if the bond has an embedded option, you will want to use effective convexity rather than modified convexity. Then the last two are also in expectation, depending on your default expectations (there's always some positive probability, no matter how small) and any expected exchange rate changes.

To summarize: [[summary]]

Neither one

The coupon income

The rolldown return

Nothing can be said with any certainty

There's a one-year forward rate of about 3% implied

There's a one-year forward rate of about 2.75% implied

It should fall

It should rise

No

Yes

Continue

Continue

Continue