Spot and Forward Rates Question

It is now January 1, 20x7. The one-year spot rate now is exactly equal to the one-year forward rate for a loan in one year as of January 1, 20x6. The current forward price of $1 par, zero-coupon bond for delivery on January 1, 20x8 will most likely be: A) the same as it was on January 1, 20x6. B) lower than it was on January 1, 20x6. C) higher than it was on January 1, 20x6. The correct answer is A. The explanation is If the spot rates evolve exactly as indicated by the forward curve, the forward price would remain unchanged.

Can anybody please explain this concept. I didn’t get the question and answer. The forward rate from 2016 to 2017 must be a straight line because the spot rate equaled the forward rate of 2016 but does that mean the curve won’t change in future?

Thanks in advance.

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It is from 2017 to 2018 but established (derived from spot curve) at the beginning of 2016, F12:

2016__ S1__2017___ F12__2018
___________S2 ______________

At 01.01.2016 Forward price of zero is 1/(1+F12). At the 01.01.2017 new one-year spot price S* 1 = F12 as stated in the problem. Forward price of 1-year zero is current spot price, because in one year zero goes to maturity: P =1/(1+S* 1). But S*1 = F12, then forward price should remain unchanged.

Think of the forward rate for a given period as the market’s anticipation of what will be the spot rate for that same period (i.e. once that period is no long forward starting).

All else being equal, the market assumes that in 1y time, the 1y spot rate will be equal to the current 1y x 1y forward rate.

Let’s take an example (similar to your example above):

  • as of March 10 2023, the 1y spot rate (2023-03-10 to 2024-03-10) is equal to R
  • as of March 10 2023, the 1y x 1y forward rate (2024-03-10 to 2025-03-10) is also equal to R (i.e. the rate curve is flat)
  • as of March 10 2023, the fair price for a 1y ZC bond is P (i.e. P is the fair price given the 1y spot rate R)

Now I ask you: as of today March 10 2023, what is the fair price for a transaction to be settled on March 10 2024 on a ZC bond maturing on March 10 2025? The answer is P, because the 1y x 1y rate is equal to R. I.e. imagine yourself moving yourself forward in time by one year and calculating the price for 1y ZC given a rate of R, your math will be the same as it is today.

Thank you both of you for your time and effort. But sorry I still didn’t get it.

This is known as pure expectations theory, one of many theories proposed to explain the shape of the yield curve.

As far as I know, it hasn’t proven to be particularly reliable in practice.

It’s a poorly worded question, and the explanation of the answer doesn’t really explain anything.

They don’t tell us the maturity of the bonds in question, nor the spot prices on 20x6/01/01 and on 20x7/01/01.

The fact that one rate evolved as indicated by the forward curve doesn’t tell us whether any other rates evolved as indicated by the forward curve, yet they’re assuming that in their answer.

I suggest ignoring it; there’s not enough information to be able to give you an explanation of what they meant.

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I was a market maker of Short Term Interest Rates (STIR) for 10 years, and I had to read the question 4-5 times before I understood what they meant / what they were after.

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Thank you very much. I will pretend like I never came across such a question :slight_smile:

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