Par Yield Curve Construction

Ask what "the 5-year interest rate" is, and the cleanest market-based answer is often: whatever coupon rate a brand-new 5-year bond would need in order to sell at exactly $1,000, its face value. That single number — a hypothetical bond's fair coupon — is the par yield at 5 years, and stringing these together across maturities gives the par yield curve, one of the most directly observable ways to describe the term structure of interest rates.

Defining par yield

By definition, a par bond satisfies:

$$100 = \sum_{t=1}^{n} \frac{c_n}{(1+z_t)^t} + \frac{100}{(1+z_n)^n}$$

In words: a bond with coupon rate $c_n$ (the par yield for maturity $n$) and face value 100 must equal 100 when its cash flows are discounted using the zero-coupon rates $z_t$ that actually apply to each individual cash flow date. Rearranged, this gives a formula for the par coupon directly in terms of the zero curve:

$$c_n = \frac{1 - d_n}{\sum_{t=1}^{n} d_t}, \qquad d_t = \frac{1}{(1+z_t)^t}$$

In words: the fair par coupon is the "missing" discount factor at maturity ($1 - d_n$, since the bond must return exactly 100 par-for-par at redemption) divided by the sum of every discount factor along the way — the coupon that exactly balances the books given the zero curve.

Bootstrapping the other direction

In practice, markets quote par yields more easily than zero rates (from on-the-run Treasury auctions, for instance), so curve builders often run the relationship backward: take the observed par curve and bootstrap the zero curve out of it, one maturity at a time, using each newly derived zero rate to solve for the next.

Worked example

A 1-year par bond yields 3.00% (so its zero rate equals its par yield exactly at 1 year, since there's only one cash flow). A 2-year par bond has a coupon of 3.40% and trades at par ($100).

1. 1-year zero rate: $z_1 = 3.00\%$ directly, since a 1-year bond has a single cash flow.
2. Set up the 2-year par condition: $100 = \frac{3.40}{1.03} + \frac{103.40}{(1+z_2)^2}$.
3. Solve the coupon leg: $\frac{3.40}{1.03} = 3.30$, so $\frac{103.40}{(1+z_2)^2} = 100 - 3.30 = 96.70$.
4. Solve for $z_2$: $(1+z_2)^2 = 103.40 / 96.70 = 1.0693$, so $z_2 = \sqrt{1.0693} - 1 \approx 3.41\%$.

The 2-year zero rate (3.41%) comes out just slightly above the 2-year par yield (3.40%) — exactly the small upward adjustment you'd expect from an upward-sloping curve.

What this means in practice

Par yields are what most people mean casually when they say "the 5-year Treasury yield" — they're what auctions target and what headline yield curve charts usually plot. But zero rates, not par yields, are the correct discount rates for pricing any bond with a coupon different from the par coupon, or any derivative cash flow. Every serious pricing system converts the quoted par curve into a zero curve first.