Don't be an id-"iota"

The easy math behind a difficult concept

In Italian (as many of you know I'm Italian) idiota means a person who doesn't understand much, while in the Greek alphabet, Iota means a small amount.

In the next few minutes I will try to explain the concepts behind the RV of bonds and specifically of bond inflation linkers, intentionally focusing on a practical and less scientific point of view.

So in the next few lines I will use the concepts of spread, Z-spread and ASW interchangeably to make it easy to explain the concepts. But I still refer you to a good volume on quantitative finance such as Wilmott or Hull for more details. Before going on remember to follow me and sharing the piece if you like it.

So let’s start.

Let's start by taking a bond linker and a nominal one with a more or less similar maturity. I used Spanish bonds which have a maturity of around 4 years. By taking the "real" returns for one and "nominal" for the other and making a simple subtraction we arrive at the so-called breakeven, i.e. the implicit inflation priced by the market over a 4-year horizon.

We can do the same thing using derivatives, taking the nominal 4-year swap (we approximate the exact maturity here, even if not exactly coinciding) and the inflation one. I put the Bloomberg tickers of both. Since for derivatives the instruments traded are the nominal and the one with implicit inflation, by difference we find the real swap rate.

Now a little methodological note, Bloomerg calls the swap with implied inflation (similar to cash breakeven) as "EUR Inflation Swap Zero Coupon Ex Tobacco", while the ticker with the real rate is called "Europe Swap Breakeven 4 Year". Don't ask me why, trust me.

Now that we have both implied inflations, we can calculate the iota as the difference of the two. We can see it in the yellow square on the right.

Using a presentation from the US Treasury, which also gives us a meaning of this indicator, this explains in an easy way that the iota is in fact the difference between breakeven on cash securities and implicit inflation of swaps (which are traded directly on the market).

A second calculation method, which will help to understand the functioning and use of inflation swaps, is through the spreads (which I have placed in the left part of the first image).

Below I plotted the yield of nominal spanish bonds vs the swap nominal curve. With an arrow I indicated our nominal bond (october 2027), basically flat vs the swap curve. The difference between the two is the Z-spread or the ASW spread (the difference between the two is that Z assume constant credit spread, while ASW uses market).

Unfortunately we cannot compare the real bond linker rate with the real swap rate, as the latter is not listed and traded. But we can transform an inflation linker bond (with variable coupons linked to future inflation) into a cash flow linked to the short-term variable rate (such as 6-month Euribor) plus a spread. This is what many companies that have positive or negative flows linked to inflation do to cover their risk and transform everything into a variable rate instrument without interest rate or inflation risk.

This is exactly we can have using the ASW function on Bloomberg. We will pay the inflation (real coupon + inflation leg) and we will receive the euribor 6M + 32bp of swap (this is thw ASW of the linker bond, basically a linker bond tranformed in fixed using the expectation of the inflation swap market compared to the nominal swap). Now calculating the difference in ASW between the nominal bond vs the ASW of linker bond (in yellow on the first chart) you can have a second version of IOTA. Clearly in this example the two are slightly different given the bonds have a maturity of (october and november) and I used the 4y swap rates. If you use interpolation for all for all steps, the two methods should converge.

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