Why a Bond Forward Price Does Not Pull to Par
Summary
The document examines how pull to par affects a bond's spot price and forward price when yields remain unchanged. The question relates the spot price, accrued interest, and financing to a fixed-date forward price, then asks how a long cheapest-to-deliver bond and short futures basis position changes over a day when net basis starts at zero.
The answer says the forward price for the same delivery date stays unchanged if the forward yield is constant; it is the spot bond that moves toward that forward price as time passes. A zero-coupon example on a flat yield curve illustrates the distinction between keeping the original delivery date and recalculating a forward for a new date one year from the later valuation date. The explanation is limited to the stated unchanged-yield setup. It does not fully work through the basis position's daily profit and loss or account for changing yields and other market effects.
Key ideas
- With unchanged yields, a bond's spot price moves toward its forward price as time passes.
- The forward price for a fixed delivery date does not itself pull to par under the stated assumption.
- A forward price recalculated for a later delivery date can differ from the price for the original date.
- The example uses a zero-coupon bond and a flat yield curve.
- The answer does not fully calculate the daily profit and loss of the futures basis position.
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Full text
# Is the Forward Price of a bond subject to the Pull to Par?
# Is the Forward Price of a bond subject to the Pull to Par?
From my understanding:
FwdPx= SpotPx - Accruals + Financing
Assume that the yield curve is flat/or that the bond yield stays the same the next day, i.e. that the market is unchanged and that the only force the bond is subject to is the Pull to Par.
Then if it is trading above 100 its price the next day will by slightly lower and the opposite if it is trading below 100. Therefore we in T+1 you actually have a change in price (call it “dP”)
My assumption is that the initial FwdPx (I.e. the FwdPx with the same starting date and end date of the previous day, not shifted by one day), will also have to move by dP + (1day Accrual - 1day Financing), due to the changes in both the spot price and the fact that the “carry” component is one day less. Is this correct?
The original question is to understand what would happen to a Basis position (long CTD, short Future), from T to T+1 if nothing at all happens, or in other words if only the deterministic components of the bond make their effect on the bond, assuming that there is a net basis equal to zero.
## Answer by dm63 (score 4)
https://quant.stackexchange.com/a/77972
No the forward price of a bond on a fixed date does not pull to par. If the forward yield stays the same, so does the forward price. In a scenario where yields don’t move , it is the spot price that gradually moves toward the forward price. Does it answer your question ?
Edit to address some comments: Consider the following example : a completely flat yield curve, all yields are 5%. A 5 year zero coupon bond has a spot price of $ 100/1.05^5 $. The forward price of this bond one year from now is $ 100/1.05^4 $. Now allow one day to pass. Assuming all yields are still at 5%, the spot price of the bond will increase slightly through the P2P to $ 100/1.05^{4+364/365} $. The forward price to the same exact date as before doesn’t change , but the forward price for the date which is one year from the new spot date increases to $ 100/1.05^{3+364/365} $.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.