# Why does the ultra long-end of a yield curve invert?

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?

• convexity is not a qualitative property: it is quantitative. An ultra long swap/bond has more convexity than a 30y swap/bond because mathematically that is a fact. Convexity is not the only driver of the shape of the long end curve. There are multiple drivers and hence the price fluctuates; with much more fluctuation if solely convexity was the only explainer. – Attack68 Aug 8 '19 at 18:56
• I think the answers are given below, in GBP above 30y market is driven by institutional investors, particularly pension funds and insurance companies. – Daneel Olivaw Aug 9 '19 at 9:24

It's an interesting question. The fundamentally devout macro wannabe-strategist within cries out for a long-term growth/inflation expectation narrative. However, the cynical realist within reminds that although the market does make long-term predictions thus because it has to create prices then, there is no latent consensus that the world will really look so different in the decade before 2049 versus after then. Or at least, it has no realistic basis for placing that mythical turning point in 2049 vs eg 2044 or 2054! And most market participants have no incentive to make that call, because they will be probably dead or almost certainly retired by then!

I suspect a much more prosaic explanation. Who actually uses 30+ year swaps, bonds, and/or linkers? Cui bono? It's pension funds, where the discount rate on their liabilities can and does move fund balances as much as near-term investment performance on the asset side. Albeit the precise mechanics differ from country to country, but the general meme is to discount incomes growing with inflation by long-term interest rates. Therefore the long-end becomes a natural hedge to pension liabilities. This is the underpinning of the ALM product suite that keeps many a fund manager and broker in claret, good hotels, plus associated vices.

Ultimately long-end anomalies are a function of regulatory fiat. Ordinarily, arbs would lean against these kinks, because they know that the market can't really credibly differentiate future outcomes two vs three or four decades out. They don't/can't because they know that someone else will come out and play the same game just as strongly, for reasons that are more legally than economically optimal.

I'm not a fixed income guy myself. But the bond investors and macro investors I do know are quite content as a group not to get too excited by the ultra-long-end, seeing it as an annex to the real market (more 5-10y tenors) borne of its own technicals as the actuaries' playground.

hope this helps.

• I've accepted this as the answer. You provide a clear explanation in real terms (i.e. real life market participants) that I interpret as predominant LDI buying/receiving in the ultra-long end keeps it suppressed, thus creating the extra convexity. Curves without LDI-presence do not exhibit this property. – quanty Jan 8 at 21:56
• Yes, exactly. But note this process is pro-cyclical. Falling long-end yields/rates require yet more ALM/LDI hedging. And vice versa. If they ever start to rise for more fundamental or issuance-vs-demand reasons, the whole process goes, and spirals, into reverse the other way. – demully Jan 11 at 20:53
• Ah yes, of course its pro-cyclical, that makes a lot of sense. It actually explains a phonemonen I observed when I first started trading GBP rates mid-last year whereby the ultra long-end of the swap curve gained a positive slope and just kept on going in quite an aggressive manner, which was then quite surprising to me. – quanty Jan 11 at 21:08

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.

• Other answers kind of hint at it, but this one is the clearest - if the curve wan't inverted at the ultra-long end then you could make put on carry, positive convexity trades and nearly always make a profit. – Chris Taylor Aug 9 '19 at 10:09
• This argument should work with 10s30s as well as 30s50s, no? In general the potential value of the roll-down (carry) and (under-priced) convexity should also be taken under advisement in relation to recent (or opportunistic) price levels and overall value at risk (delta) of the trade. – Attack68 Aug 9 '19 at 14:14
• @attack68 If you do this trade with 10s30s what you find is that 10s tend to move more than 30s, which offsets the effect. Whereas the market tends to move 30s and 40s in parallel. – dm63 Aug 10 '19 at 2:59
• Thanks @dm63 Would you mind posting thats maths that make it a mathematical fact? – quanty Aug 11 '19 at 8:59
• Well to simplify, consider only zero coupon bonds with a yield y and maturity T: $Price, P=e^{-yT}, dv01 = dP/dy = -Te^{-yT} and convexity = d^2P/dy^2 = T^2 e^{-yT}$. Hence the convexity per unit dv01 is proportional to T. – dm63 Aug 11 '19 at 13:22

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.