Why a Par Bond Coupon Matches the Par Swap Rate

Summary

The document explains why the coupon rate on a bond issued at par matches the fixed rate on a par swap with the same payment schedule and maturity. In the example, a two-year bond pays semiannual coupons, and the question compares its par coupon with the quoted two-year swap rate after discounting the cash flows.

The explanation rests on matching payment streams: when the bond is priced at par, its fixed coupon payments and principal repayment have a present value equal to par. Those fixed payments correspond to the fixed leg of the swap, so the rate that makes the swap's fixed leg fair is the bond's par rate. The excerpt gives an intuitive equivalence rather than a full derivation, and it does not discuss conventions such as day counts, payment calendars, or differences in instrument terms that can affect real market comparisons.

Key ideas

• A par bond's coupon payments and principal have a present value equal to its par value.
• The fixed payments on a bond can match the payment schedule of a swap's fixed leg.
• With matching maturity and payment terms, the par bond rate equals the par swap rate.
• Market convention differences can matter when comparing actual bonds and swaps.

Tags

# Swap Rate vs Par Rate


# Swap Rate vs Par Rate












If we calculate the par rate for n periods, why does the nth swap rate equal the par rate? A mathematical formulation would be helpful apart from an intuitive answer.

> Edit: Example:- A 2 year bond pays semiannual coupons and has a par value of $100. Swap Rates- 0.65% (0.5year), 0.8% (1year), 1.02% (1.5 years), 1.16% (2years). Computing discount rates from this gives- 0.9968, 0.9920, 0.9848, 0.977 respectively. Now the par rate would be - $Par/2 * [(d(0.5)+d(1.0)+d(1.5)+d(1.5)+d(2.0)] + 100 * d(2.0) = 100$ Gives par = 1.16%

## Answer by wsw (score 3, accepted)

https://quant.stackexchange.com/a/21349

Since your 2-year bond is at par, the fixed coupon payments over the 2 years match the payments in the fixed leg of the 2-year swap exactly. Hence the par rate of the bond is the same as the par swap rate.

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.