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The shape of the yield curve (at least in the GBP Rates market) is upward sloping from the front end up to the long end (i.e. 30y), but then begins to become downward sloping as we go beyond 30y and 40y. (Although, at the time of writing, and I think for the first time ever, the 30s50s curve has become upward sloping.)

I know that the reason for this is: convexity.

However, I'm not entirely sure why the ultra-long end has this extra convexity, and why it is necessary that the extra convexity implies an inversion of the slope.

My thoughts

The extra convexity: In a normal scenario, as we go further out of the curve, our certainty about what kind of interest rate regime we might be in reduces, meaning that the long end has to be flatter than the front end. In order to maintain this flatness, there has to be a point of reasonably high convexity as the front end morphs into the long end. (I.e. steepness manifesting as reasonable expectations morphs into 'who-knows-what-rates-will-be-in-20-plus-years'.) It's reasonable to have a view on where rates will be in 1 to 5 years, maybe even 10, but beyond that you can pretty much forget about it, so there is no need for significant steepness.

But why do we see a change in convexity from 30y+? Why does the curve not just level out beyond 30y? Why does it dip lower?

I would not say that this is universally acknowledged but here is my view:

Instead of considering par rates, i.e. 10Y and 20Y, consider forward rates, such as 10y and 10y10y. The useful difference here is that forwards do not 'overlap' and therefore incorporate aspects of each other into the price. A 20Y is >50% directly dependent upon the 10Y price for example.

Consider the approximate values of convexity for a $ 1000 PV01 delta IR swap:

• 10Y: $ 1.1


• 10Y10Y: $ 3.1


• 20Y10Y: $ 5.1


• 30Y10Y: $ 7.1


• 40Y10Y: $ 9.1

Now what is the value of gamma (convexity)? Well that depends on the volatility of the rates. If all rates are assumed to have the same volatility, e.g. 50bps per annum then the expected value of each of these over one year is calculated as:

$$ 0.5 times 50^2 times gamma = text[1.4bps, 3.9bps, 6.4bps, 8.9bps, 11.4bps]$$

And that is only for one year, even though these are swaps with a longer tenor.

There are other potential factors that are acknowledged if volatilities are not consistent across the curve, which not only impacts the value of convexities but also affects the mean expectation of where rates will be under a log-normal assumed distribution of prices.

I would recommend chapter 8 of Darbyshire: Pricing and Trading Interest Rate Derivatives which also references Litterman-Scheinkman-Weiss: Volatility and the Yield Curve in its discussion.

Suppose 40yr bond and 30yr bond have the same yield. It is a mathematical fact as @attack68 has pointed out, that the convexity of the 40yr is greater than the convexity of the 30yr bond. So consider the following strategy ; long the 40 yr bond and short the 30yr bond with the same dv01. Then every time the market moves, you make money (get longer when the mkt rallies and shorter when it sells off). The market does not give this for free, so it charges you by making the 40yr bond yield lower.

The 30yr and greater is really a product for insurance companies and sovereigns. Insurance companies dominate the swap and futures market there and are the biggest real money players with hedge funds and dealers typically taking the other side of those trades. This is especially the case in swaptions and exotic structures out there as well. Equity returns and hedging of those instruments drive a lot of the real buying or selling volume there and dominate the real money flow.