# How does the term premium of the 10y20y Treasury forward rate relate to the 30y rate?

I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate (neutral rate).

It then says that, as the term premium of the 10y20y Treasury rate is unlikely to become higher due to certain factors such as the term premium being high enough to make up for expected supply, there is not much of a risk in a sell-off in the 30y Treasury.

Please can someone explain to me how the 10y20y forward rates will end up affecting the 30y rates? In intuitive terms would be appreciated.

As well as the above, can someone please expand on how the Treasury supply affects the forward rate?

Hmm...

I have, in a past life, been down to Liberty Street to "consult" with the Markets Division on asset pricing. For those maybe not au fait with the jargon, any interest rate can be broken down into parts. So a 30 bond yield can be broken into a 10y yield and a 10y20y yield (ie a 20 year rate of interest starting in 10 years time, that follows the yield received in the first 10 years). Together, these will compound to the 30y yield over the whole life of the bond. If I know what the first 10 year yield is, the next 20 years are a derivative of this. Which effectively allows me to create a series of forward interest rates for any period of time in the future.

So I could pay 9 year rates and receive 10 year rates, locking in the 9y1y rate (ie 1 year rate starting in 9 years time). This will typically (historically have been) higher than the actual 1 year rate then, which is the "term premium" referenced in the question. It's simple time-value-of-money, rather than arbitrage. Every reason I would require a risk premium for buying 10y paper over 1y paper says this should be in the 9y1y almost as much as in the 10y versus the 1y! And if it wasn't, that would create an arb.

So I'm wondering how your guys are measuring their "long run nominal fed funds" here ;-) This baseline, from which TP is measured, is the great unknowable but essential variable here...

ps so imagine a couponless 5 year rate at 2%. TR = 1.02^5 = 1.1041. Meanwhile, a couponless 10 year rate is 2.5%. TR = 1.025^10 = 1.2801. So the TR on the 5y5y = 1.2801/1.1041 = 1.5941. 1.5941^0.2 = 1.0300, so that's a 5y5y rate priced at 3.00%. The same applies with respect to your 10y20y versus 30y and 10y.

• Thanks. I think the long run nominal funds rate is being defined as the neutral rate. Is that any easier to answer? Also, your examples make sense, but if byou pay the 9 year rate and receive the 10y rate you end up receiving 10y - 9y, which isn't exactly the 9y1y rate, no? doesn't it just become a first order approximation of the 9y1y rate? – junior_pm Jan 23 at 1:54
• Two supplementary questions there... if the long-run short-term nominal is "neutral", then is your "r-star" a Laubach-Williams, a Holstein-Laubach-Williams, or any variations of the Taylor Rule, steered to your subjective prejudices about the future? ;-) See newyorkfed.org/research/policy/rstar - I'm not feeling their love for rates pricing being close to equilibrium expectations ;-) – demully Jan 23 at 2:11
• I wish I had an answer to that! – junior_pm Jan 23 at 2:13
• On the second one = 10y - 9y = you are strictly correct (sort of). It's neither exactly the same as 9y1y... but neither is it the same as 10y-9y (given the duration mismatch). Either of us could be correct IF the two legs were correctly sized (for our perspective on this). I was speaking figuratively; and should have been specific to be correct. The same could be said for equal quantities of the 10y versus the 9y mis-hedging the 10y-9y spread! Devil in detail, here. Hopefully, the concept of message intended clear, and not problematic, on either front. – demully Jan 23 at 2:25
• Ah of course - duration mismatch - I agree. Do you mind taking a look at my question on how supply affects forward rates as well? Sorry for bombarding you with questions – junior_pm Jan 23 at 2:27

In swap space the 20y rate 10y forward (10y20y) is related to the 30y rate 0y forward (0y30y or just 30y) by the equation:

$$R_{10y20y} = \frac{x}{z}R_{30y} - \frac{y}{z} R_{10y}$$

(where x,y,z are day fraction and discount factor scalars)

Treasury rates have a similar formula albeit uses a geometric structure rather than arithmetic due to how bond yields are expressed.

But that is besides the point. The key is that since 10y20y is directly dependent upon 30y and 10y rates if the 30y rate goes up (10y staying the same) then the forward must go up also, as a mathematical dependent.

• Thanks, but I am curious to know how the term premium itself of the forward, as mentioned in the post, affects the 30y rates. Do you mind answering this? – junior_pm Jan 22 at 14:23
• what is the term premium? who has defined it, and as what? but regardless of that if the 10y20y rate is somehow composed of "base_rate + term_premium", and that the "term_premium" component is somehow maxed out then the only way for 10y20y to go higher is for the "base_rate" component to go higher which obviously limits the number of ways by which you author is allowing the 10y20y rate to go higher. Then they assert that the 30y must be similarly constrained due to its mathematical relationship with 10y20y. – Attack68 Jan 22 at 14:33
• By term premium they have defined as the premium over the long run nominal funds rate (it is in my post). Do you also mind explaining how the Treasury supply affects the term premium of the forward rate? – junior_pm Jan 22 at 14:40
• so the 'term_premium = current rate - long run nominal funds rates', and your author is suggesting because the term premium is above zero this is high and therefore wont go higher. Basically all the author is saying is that the current rate (of 10y20y) should be at or below the long run nominal funds rates. Im sorry but this is all so vague and so much waffle that I cannot, in good conscience, decipher it. Not all research is always good. More supply means propensity for higher rates, but higher rates means more propensity for demand, i.e. greater supply can be absorbed at high rates. – Attack68 Jan 22 at 14:56
• Basically the research says that there is a 25bp premium to the nominal funds rate, and due to the fact that the term premium is enough to 'compensate' for any supply and factors like contained volatility, which provide reasons for lower term premia, 30y Treasuries shouldn't drop in price. Does this make it any clearer? – junior_pm Jan 22 at 15:07

"can someone please expand on how the Treasury supply affects the forward rate?"

All else equal, more supply should mean lower bond prices (higher bond yields) as Attack68 noted. However, I think the Street has historically assumed that supply will affect spreads (e.g. swap spreads) more than the level of rates.

• Sure, I get that more supply should mean lower bond prices, but how does this work for forwards? – junior_pm Jan 23 at 2:03
• "how does this work for forwards? " --- See @demully comment "30 [year] bond yield can be broken into a 10y yield and a 10y20y yield". – user42108 Jan 23 at 15:49
• I mean, intuitively, would it be the supply of the 30y bond that affects the 10y20y yield? I don't think I quite get it – junior_pm Jan 23 at 16:11
• The US ended the sale of 30y bonds in late Oct 2001. Perhaps you want to look at the impact on spot and forward yields around that date. It was a negative supply shock and is as close to a 'natural experiment' as you're going to get. Might help with understanding the impact of supply on yields. HTH. – user42108 Jan 26 at 18:48
• reuters.com/article/usa-bonds-idUSL1N2JO2GV I thought they are still selling 30y bonds?! – junior_pm Jan 28 at 18:58