Par Yield Curve

Par Yield Curve: Definition, Meaning, and How to Derive It

What is the par yield curve?

The par yield curve plots the coupon rates (yields to maturity) of hypothetical coupon‑paying Treasury securities that would be priced at par (price = 100). For each maturity on the x‑axis, the par yield is the annual coupon rate that makes the bond trade at par given prevailing discount rates. The curve offers a complementary view to the spot and forward yield curves and is useful for pricing new coupon bonds and comparing market interest‑rate expectations across maturities.

How the par curve differs from spot and forward curves

• Spot yield curve: shows zero‑coupon (spot) rates for different maturities. These rates are used to discount individual cash flows.
• Forward curve: shows implied future short‑term rates derived from the spot curve.
• Par yield curve: gives the coupon rate that makes a coupon bond’s price equal to par. It aggregates multiple spot rates to express the single coupon level consistent with current discount factors.

Under a normal upward‑sloping interest environment, the par yield curve typically lies below the spot curve for intermediate maturities, because the par rate is a weighted average of several spot rates (one for each coupon date).

Why the par curve matters

• Determines the coupon rate a new bond must carry to issue at par.
• Provides a market‑based summary of yields on coupon bonds (as opposed to zero‑coupon yields).
• Serves as an input or cross‑check when bootstrapping the spot curve and deriving forward rates.

Deriving the par yield curve (bootstrapping overview)

Bootstrapping is the standard method to derive spot (zero) rates from observed par yields. The idea:
1. Use the shortest-maturity par yield (often a zero‑coupon instrument) to obtain the shortest spot rate directly.
2. Move to the next maturity: set the present value of a hypothetical par bond’s cash flows (coupons and principal) equal to par, discounting known coupon payments with previously derived spot rates and solving for the unknown spot rate for the final payment date.
3. Repeat iteratively to produce spot rates for successively longer maturities.
4. From the complete set of spot rates you can compute forward rates if needed.

Simple numerical example (assumptions: semiannual coupons)

Assume face value = $100. Par yields (annual) observed in the market:
* 6‑month par yield = 2.00% (annual, so semiannual coupon = 100(0.02/2))
* 1‑year par yield = 2.30% (annual, so semiannual coupon = 100(0.023/2) = $1.15)

Step 1 — six‑month spot rate:
A six‑month par bond with one payment trades at par, so the six‑month spot rate is 2.00% (annualized).

Step 2 — solve for the one‑year spot rate:
Price at par (100) equals the present value of two cash flows:
100 = 1.15 / (1 + 0.02/2) + 101.15 / (1 + s1/2)^2

Where:
* First term discounts the first semiannual coupon using the six‑month spot rate (2.00% annual).
* Second term discounts the second coupon plus principal using the unknown one‑year spot rate s1 (annual, semiannual compounding).

Solving yields s1 ≈ 2.302% (the one‑year spot rate). The same approach extends to longer maturities by using already‑derived spot rates to discount earlier coupon payments and solving for the next unknown spot rate.

Practical notes

• Par yields are usually quoted on an annual basis with standard market day‑count and compounding conventions (confirm conventions when computing).
• Bootstrapping requires a consistent set of market par yields across maturities. When market data are missing you may need interpolation or curve‑fitting methods.
• The par curve is especially useful for market participants focused on coupon bonds, while the spot and forward curves are more natural for discounting and modeling interest‑rate dynamics.

Key takeaways

• The par yield curve shows coupon rates that make coupon bonds trade at par for each maturity.
• It is effectively a weighted composite of spot rates and differs from the spot and forward curves.
• Bootstrapping converts observed par yields into a complete spot curve by solving sequentially for zero‑coupon rates.
• Understanding the par curve helps with bond issuance, pricing, and interpreting interest‑rate expectations.