How Treasury Forward Rates Connect to Long-Term Yields and Term Premium

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


# 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?

## Answer by demully (score 3)

https://quant.stackexchange.com/a/60642

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...

Scratching forehead here, DEM

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.

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/60629

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.

## Answer by user42108 (score 0)

https://quant.stackexchange.com/a/60637

"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.