I hope you enjoyed the previous two instalments of these paper teach-ins. I wouldn’t blame you if it were too hard to get your head around it. If you, however, went down the road of creating it all in a spreadsheet, you should have cracked the terminologies and understanding, hopefully. If not, no worries; take your time, and it will eventually click.

Now, let’s go into the weeds of yield curves themselves. Open up the spreadsheet you might have created, as it will help you in this next session. It is basically following the below table. You’ll understand more if you calculate columns B, C and D yourself. The link to the document is here.

Similar to the script, it is worth homing in on the notion that if forward rates materialise, all bonds in the above table will earn exactly the same 1-year return. Similarly, positively-carrying bonds (higher yield advantage versus 1-year spot) will have to suffer capital losses, while negatively-carrying positions are expected to experience capital gains.

The example here is to analyse a 3-5s yield curve flattening positioning. The current yield gap of this trade is (8.73% - 7.75%) = 0.98% (98 bps). An upward-sloping yield curve (typical shape) is normally carrying negatively. Also, yield curve trades are duration neutral, meaning that (in theory) they are immune to parallel interest changes. In order for this trade to be duration neutral, we are selling 100% of the 3-year spot and buying 50% of the 5-year bond and 50% of the 1-year bond. Duration wise we are selling three years’ worth of duration but buying 2.5 (50% of 5) and 0.5 (50% of 1), which equates to zero: -3 + 2.5 + 0.5.

The carry of the position is the weighted yield of the overall package:

-100% * 7.75%

+50% * 6.00%

+50% * 8.73%

= - 0.39%, so the position is carrying negatively

What does it tell us? We should expect some flattening in the forwards to enjoying capital gains which will hopefully offset the negative carry.

How much flattening? Well, looking at the 1-year forwards for each spot rate, we can calculate what the implied horizon return for the flattening positions is:

-100% * 9.27%

+50% * 3%

+50% * 10.43%

= - 1.05%

The yield curve position must flatten by 0.52% at the horizon to be flat on this return. The initial 3-5s curve is a 2-4s curve in 1 year’s time. At horizon, it will have a duration of 2 (100% short 2-year and 50% long 4-year and 50% 0-year). Dividing the -1.05% by 2 gives you the implied flattening that needs to occur for this flattened to break even. 0.52% or 52 bps is the answer. The same conclusion could be reached when looking at the spot rates 1-year forward. From the table, we can see that the 2-year spot rate is expected to increase by 1.64% in a year’s time while the 4-year rate is set to jump by 1.11%. This implies a flattening of (1.64% -1.11%) = 0.52%.

I hope that all makes sense. A little homework for you, if I may. Why don’t you look at the above principles and calculate the curves for the following segments:

**4-7s6-10s3-8s**

remember that they all turn into a different curve in a year’s time.

I suggest creating a table like this below, where you summarize and calculate the basic statistics. Remember that the initial curve trade has to have zero duration. In my example, the weights of 50% worked nicely. You might have to move those weights for some other examples to be duration neutral. I hope you get there. I will post the answers to the above curves in a week’s time for your reference. Good Luck.

### Further Reading

If you are interested in further reading and more detailed work, I suggest you go after the remaining documents here:

Does Duration Extension Enhance Long-Term Expected Returns"?

Convexity Bias and the Yield Curve

A Framework for Analyzing Yield Curve Trades

Enjoy!

### Music

I got a soft spot for acoustic covers. I came across this: taking Linkin’ Park’s “Numb” and propelling it in a different direction. It even chills out any dog instantly. Enjoy it and have a chilled week ahead!

Your,

Paper Alfa

Great series! I’m still working to understand it all hahah but i’m having an odd problem where when i download the linked papers when i zoom in the text goes blank.

thank u