I hope you immersed yourself in the Antti Ilmanen yield curve concept I mentioned in my previous post. If you haven’t done already, I suggest you download it here. It’s not the easiest read especially if you are just new to bond yields and curves, but believe me, if you understand those concepts, you are ahead of the pack and, in some cases, many PMs I have come across over the years.

Once you have familiarised yourself with this concept, I suggest downloading part two of the series here. Now, in this section, he goes one step deeper into the forward curve and yield curve analysis. I touched on some of the items in Yield Curve - 1, but I will review them again while referencing parts of the excellent paper.

The best way to learn this is to open a spreadsheet and try re-creating *figure 1*.

You can copy and paste special (as text) the following dump into a spreadsheet as a start:

`Maturity Spot 1y Fwd Rate Impl Spot 1y Impl Chg in Spot`

`1 6.00% 6.00% 8.01% 2.01%`

`2 7.00% 8.01% 8.64% 1.64%`

`3 7.75% 9.27% 9.09% 1.34%`

`4 8.31% 10.01% 9.42% 1.11%`

`5 8.73% 10.43% 9.67% 0.94%`

`6 9.05% 10.66% 9.85% 0.80%`

`7 9.29% 10.74% 9.97% 0.68%`

`8 9.47% 10.74% 10.06% 0.59%`

`9 9.60% 10.65% 10.12% 0.52%`

`10 9.70% 10.60% `

It’s best just to have the spot rates, to begin with, and try populating the rest. 1y Fwd Rates are calculated using the corresponding spot rates. For example, the 1-year rate 1-year forward is calculated using 1-year and 2-year spot rates. More precisely, by doing the following calculation:

(1.07)^2/ (1.06)^1 -1 = 8.01%

You can then basically copy the formula down for all different maturities. This is the market’s implied path of 1-year rates over the next ten years.

Column C is trying to calculate the spot rate but 1-year forward. In other words, what is the yield curve implying for spot rates to be 1-year forward.

Now for 1-year rates, we have already done the calculation and know that it’s expected to rise from 6% to 8.01%. What about the 2-year rate?

You can do this two ways, really.

Using 1-year and 3-year spot rates, the calculation is as follows:

((1.0775)^3 / 1.06)^ (0.5) (or square root) -1 = 8.635%taking the square root basically annualises a 2-year rate

You can, however, also construct the same rate by compounding the 1-year rate in 1-years’ time with the 1-year rate in years’ time. The calculation for this would be:

((1.08^1)* (1.0927)^0.5) +1 = 8.635%

Finally, column D is the change of the Spot rate in 1-years’ time to the current spot rate. It signifies how much yields have to change to offset the carry and therefore equal the horizon return for all bonds.

For example. The 2-year spot rate would have to increase by 2.01% from 7% to 9.01% to offset its carry position. Why is that? They yield advantage versus the 6% 1-year bond is 1%. This would be lost by an increase of 2.01% due to capital losses. 1%, however, would be recouped as the 2-year rolls down to become a 1-year. So, overall it would lose 1% due to higher rates + roll-down effect, thereby equally the 6% return of the 1-year bond.

Familiarise yourself with those concepts first. It will shape all of your understanding when we tackle yield curve or barbell positions next.

### Add on: (Solution)

Hopefully, that will help you compare results in your own spreadsheet.

### Music

I had the pleasure of seeing Damien Jurado live last year in a small Hackney club on a wet Sunday evening. His melancholic tunes just reassure me that, for some reason, everything will always be fine. Strange, isn’t it?

You might not have heard of him, but that dude can roll and has published numerous albums. Some songs, like the one below, are just absolute classics, in my opinion.

Anyway, I hope you enjoy it! And now, get cranking on that spreadsheet!

Your

Paper Alfa

Just cant seem to get the spreadsheet to spit out the same figs you get in col C - Im way out. I get 8.64 as in the first eg, but after that Im way out