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Why a Floating-Rate Bond’s Z-Spread Can Differ from Its Coupon Spread

Article Quant Q&A · Author: slothish1
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Summary

The document asks why a bond priced at par, with a coupon tied to SOFR plus a fixed margin, can have a calculated z-spread different from that margin when the benchmark curve is also SOFR. The example describes a five-month bond with monthly payments and a downward-sloping set of SOFR rates; the reported z-spread exceeds the coupon margin. The question also observes that curve direction appears to affect the difference.

The answer points to a mismatch in rate interpretation: the coupon calculations use forward rates for individual accrual periods, while the comparison assumes term rates that apply from today through each payment date. Those rate conventions are not interchangeable, so the bond’s pricing equation does not necessarily return the coupon margin as its z-spread. The document gives a conceptual diagnosis rather than a full derivation or worked repricing. It does not establish a general numerical relationship for every curve or convention, so the result depends on matching the curve construction, cash flow dates, and discounting assumptions.

Key ideas

  • Forward rates for individual coupon periods differ from term rates measured from today to each payment date.
  • A z-spread depends on how benchmark rates are interpreted in discounting.
  • A floating coupon tied to a benchmark does not by itself guarantee that the z-spread equals its margin.
  • Curve shape can affect the observed difference when the rate conventions are mismatched.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# Why is z-spread of bond at par value not equal to interest rate spread over benchmark?


# Why is z-spread of bond at par value not equal to interest rate spread over benchmark?












Bond: \$1,000 outstanding principal, pays SOFR 1M + 2.00% monthly (i.e. divide by 12), matures in 5 months, is worth \$1,000 today.

If the bond's benchmark curve is the same curve as its coupon is tied to (SOFR 1M), why is the calculated z-spread not mechanically equal to the coupon spread? Based on my testing, it seems that for a decreasing curve the z-spread will be greater than the coupon spread, while for an increasing curve the z-spread will be less than the coupon spread. Using as an example 4.40%, 4.10%, 4.00%, 3.70%, and 3.30% as the periods 1-5 values of the SOFR 1M curve, I calculated z-spread as 2.76%, but I expected it to be equal to the coupon spread of 2.00%.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/82508

It appears that in column C the rates are forward rates applicable for each coupon , whereas in column I you have assumed they are term rates applicable for the whole time 0 to n.

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