Interest Rate Forward vs Spot Interest Rate

[VanBuren77]

Hi -

New to the forum, so not entirely certain of the format here. I have a question:

Forward interest rates are indicative of the market's expectation. They are weak predictors of the future, but they tell us an idea of how the market feels about interest rates in the future.

If I have a 10 year bond with a yield of 2% as of today, and a 10 year, 1 year forward bond with a yield of 3%, then I should have an upwards slowing forward rate curve. Because price and yield have an inverse relationship, the price of the 1 year forward bond should be lower than the price of the 10 year spot.

How does this make sense in the context of these rates / prices being indicative of the market's expectation of the future?

If the market expected interest rates to rise, wouldn't there be a rally in rates, such that the price of the 3% forward rate bond would rise, and offset the yield?

 

[Lu Shu Kai FRM]

Hi @VanBuren77,

Just to understand your question better:

How does this make sense in the context of these rates / prices being indicative of the market's expectation of the future?
- Are you asking how well the forward rates predict the actual rate later?

If the market expected interest rates to rise, wouldn't there be a rally in rates, such that the price of the 3% forward rate bond would rise, and offset the yield?
- I don't quite understand what you mean by "rally" in rates.

 

[VanBuren77]

Hi @lushukai -

Thanks for getting back to me.

"Are you asking how well the forward rates predict the actual rate later?"

No, I am not. Like I said above, today's forward rates are poor predictors of realized forward rates.

"I don't quite understand what you mean by "rally" in rates."

When the market rallies, it simply means the market begins to buy. If someone ever says to you on a trading desk that interest rates are rallying, it means yields are dropping because people are buying.

Hope that helps.

 

[David Harper CFA FRM]

HI @VanBuren77 See my note here https://forum.bionicturtle.com/thre...rates-if-you-have-bond-prices.4927/post-13276 where I use an old GARP question to show "F[3,4] = P(3)/(P4) - 1; i.e., the efficient answer given by GARP" but your forward bond already prices (embeds) the forward rate, but I still don't quite follow your question. To me, it just begs an illustration: when I go to illustrate with numbers, the ambiguities are immediately teased out. This looks like a true statement: "Because price and yield have an inverse relationship, the price of the 1 year forward bond should be lower than the price of the 10 year spot." ... but I have a bit of trouble interpreting this statement, and a simple illustration would surely resolve: "If the market expected interest rates to rise, wouldn't there be a rally in rates, such that the price of the 3% forward rate bond would rise [if the forward rate increased, which would we expect if rates increase, then yield of the forward bond would increase, and its price would decrease, I would thinks]".

It's all in the question formulation. The follow-on issue is that the implied forward rates may or may not be unbiased predictors of future spot rates. Thanks,

 

[VanBuren77]

@David Harper CFA FRM thanks. Rethinking this problem, let me rephrase it into something much more simple. Let's assume:

P(10) > P(F(10,1))

Under this assumption, the forward rate curve is upwards sloping. Why wouldn't the market buy the forward rate contract and offset the additional yield?

 

[David Harper CFA FRM]

I guess you mean P[F(0,10,1)]; i.e., today's price of the forward contract. But, yea, it's called a repo carry: if the term structure is upward sloping and static, then you can buy the long rate and fund it by rolling over the short-term rate. In which case, the term structure embeds a premium and the forward rates are not predicting the future spot rates. Say the spot rate term structure is 1% @ 1-year, and 2% at 2-years such that the 1-year forward rate is (c.c) 3.0%. That's the implied forward because exp(1%)*exp(3%) = exp(2%*2). If the term structure is static, then you can buy the 2-year rate (earn 2.0%) and profit by funding it with a 1-year at 1% and "repo rollover" (so to speak) at 1% in one year. You profit because the implied forward of 3.0% did not predict the actual future spot of 1.0% (because the term structure is static). The upward sloping term structure embeds a risk premium. On the other hand, if you go forward one year and the spot is 3.0%, then your carry trade does not profit.

 

[VanBuren77]

I guess you mean P[F(0,10,1)]; i.e., today's price of the forward contract. But, yea, it's called a repo carry: if the term structure is upward sloping and static, then you can buy the long rate and fund it by rolling over the short-term rate. In which case, the term structure embeds a premium and the forward rates are not predicting the future spot rates. Say the spot rate term structure is 1% @ 1-year, and 2% at 2-years such that the 1-year forward rate is (c.c) 3.0%. That's the implied forward because exp(1%)*exp(3%) = exp(2%*2). If the term structure is static, then you can buy the 2-year rate (earn 2.0%) and profit by funding it with a 1-year at 1% and "repo rollover" (so to speak) at 1% in one year. You profit because the implied forward of 3.0% did not predict the actual future spot of 1.0% (because the term structure is static). The upward sloping term structure embeds a risk premium. On the other hand, if you go forward one year and the spot is 3.0%, then your carry trade does not profit.

Seeing this listed as P[F(0, 10, 1)] makes me realize I'm not thinking of this clearly. Thanks, big help

I think a more simple framework is to think of forward rates as P[F(10, 1)] and keep it in this context