Matching Forward Rate Periods to Bond Numeraires

Summary

The document asks whether a zero-coupon bond numeraire is priced using a forward rate over the wrong accrual period. It defines each forward rate as applying to an interval between adjacent dates and gives a bond valuation as a product of discount factors across those intervals. The question then compares that convention with a quoted cost for a bond numeraire at an intermediate date, where the stated accrual interval appears inconsistent with the forward-rate index.

No resolution or answer is included, so the document does not establish whether the cited formula is a mistake or a misunderstanding. Its useful point is the need to check date indices and accrual periods consistently when constructing bond prices and numeraires. It provides a setup and a specific suspected mismatch, but no derivation, market evidence, or broader treatment of fixed-income modeling.

Key ideas

• Forward rates are defined for specific adjacent-date accrual intervals.
• A zero-coupon bond price can be expressed as a product of discount factors for the remaining intervals.
• The question concerns whether a numeraire's quoted discount factor uses the correct forward-rate period.
• The document presents the issue without resolving it.

Tags

# Mismatch of periods with numeraire compared to the forward rates


# Mismatch of periods with numeraire compared to the forward rates












In Joshi's The Concepts and Practice of Mathematical Finance Page 323--324 I believe that there may be a mismatch of periods with forward rates:

Consider time partition $t_{0} < ... < t_{n}$ where $f_{j}(t)$ is the forward rate for period $[t_{j},t_{j+1}]$ for any $j = 0,..., n-1$ at time $t$. Therefore a ZCB $B_{n}$ that matures at $t_{n}$ can be valued for $j<n$ at time $t_{j}$ as $$B_{n}(t_{j})=\prod\limits_{k = j}^{n-1}\frac{1}{1+f_{k}(t_{j})(t_{k+1}-t_{k})}.$$

Let $B_{j}$ denote the ZCB maturing at $t_{j}$ for $j = 1,...,n$, then at time $t_{j-1}$ we purchase $B_{j}$ and then at time $t_{j}$ we reinvest some of the proceeds of the matured bond into $B_{j+1}$ and we proceed as such across the timeline until $t_{n}$.

This is correct and makes sense to me. But then it is further stated that:

At time $t_{1}$, the Numeraire $B_{2}$ will cost $\frac{1}{1+f_{1}(t_{1})(t_{1}-t_{0})}(*)$.

In my view this is incorrect and should rather cost $\frac{1}{1+f_{1}(t_{1})(t_{2}-t_{1})}$.

I do not understand why the period $[t_{0},t_{1}]$ is being used in $(*)$ rather than $[t_{1},t_{2}]$, since the forward rate $f_{1}$ runs over $[t_{1},t_{2}]$ by definition. Have I found a mistake or misunderstood it?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.