When a Swap Fixed Rate Represents Par Yield

Summary

The document asks whether the fixed rate on an overnight indexed swap, or on a general fixed-for-floating swap, can be treated as a par yield when bootstrapping discount factors. It presents a relation in which discounted fixed coupons plus the final discount factor sum to the value of the notional payment. It also proposes that compatible discount factors and implied forward rates may lead to the same relationship, and asks whether this reasoning is sound.

No answer or derivation is included, so the document does not establish when the fixed rate is a par yield or validate the proposed equivalence. It is useful as a statement of the pricing question and the equations under consideration, but it cannot serve as a complete method for curve construction. Any practical application would need to account for the swap’s conventions, payment schedule, floating-rate index, and discounting framework, none of which are resolved here.

Key ideas

• The document asks whether an OIS fixed rate can be treated as a par yield for bootstrapping discount factors.
• It gives a discounted-coupon relation involving the fixed rate and the final discount factor.
• It proposes a link between discount factors and implied forward rates.
• The document contains no answer, so the proposed equivalence remains unverified.
• Applying the relation in practice requires swap conventions and curve details not supplied here.

Tags

# Is a swap fixed rate always a par yield?


# Is a swap fixed rate always a par yield?












I am learning about using the OIS fixed rate to value a plain vanilla LIBOR swap. I'm using Bond Math by Smith, and the accompanying online addendum.

To bootstrap the discount factors, the author treats the OIS fixed rate as a par yield. If $d_i$ are the OIS discount factors, $R_N$ is the OIS fixed rate, $m$ is the periodicity, then this means that for any maturity $N$ we have

(!) $\frac{R_N}{m}\sum_{i=1}^Nd_i + d_N = 1$.

My questions are:

1) Why should the OIS fixed rate be a par yield?

2) General question: Is the fixed rate of an arbitrary fixed/floating swap always a par yield in the above sense?

Here is what I suspect, based on my (limited) experience with plain vanilla LIBOR swaps. If $f_i$ are the relevant "implied" forward rates, we want

$(1+f_i/m) = \frac{d_{i-1}}{d_i}$ for all $i$.

It seems that this condition is equivalent to the requirement that

$\sum_{i=1}^Nf_id_i + d_N = 1$

and hence to equation $(!)$ above by the definition of $R_N$. In other words, equation $(!)$ is a way of enforcing the compatibility of the $f_i$ and the $d_i$ without explicitly referring to the $f_i$.

3. Is the above speculation correct, or at least on the right track?

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