What Is the Par Yield Curve?

What Is the Par Yield Curve?

Let me explain the par yield curve to you directly: it's a graphical representation of the yields on hypothetical Treasury securities that are priced at par, meaning their coupon rate matches the bond's yield to maturity. This approach allows you to evaluate how prices and yields interact in the bond market, giving you perspectives that differ from what you'd get from spot or forward yield curves.

Exploring the Dynamics of the Par Yield Curve

You know the yield curve as a graph plotting interest rates against bond yields for maturities from three-month Treasury bills to 30-year bonds, with interest rates on the y-axis and time on the x-axis. Short-term bonds usually yield less than long-term ones, so the curve slopes up to the right. When people talk about the yield curve, they often mean the spot yield curve for risk-free bonds, but sometimes they're referring to the par yield curve instead.

The par yield curve specifically graphs the yield to maturity (YTM) for coupon-paying bonds across various maturities. YTM is the return you expect if you hold the bond until it matures. When a bond issues at par, its YTM equals the coupon rate, and as rates change, YTM adjusts. If rates fall after issuance, the bond's value rises because its coupon is higher than current rates, making the coupon rate exceed the YTM in that scenario.

YTM acts as the discount rate where future cash flows equal the current price. A par yield is the coupon rate that prices the bond at par—essentially zero premium or discount. So, the par yield curve plots YTM against maturity for bonds at par, helping you figure out the coupon rate needed for a new bond of a certain maturity to sell at par today.

This curve provides a yield for discounting multiple cash flows on coupon-paying bonds, using spot yield curve data to apply the correct spot rate to each coupon. In an upward-sloping environment, the spot yield curve is above the par yield curve, and the reverse holds in downward slopes.

How to Derive the Par Yield Curve: A Step-by-Step Guide

Deriving the par yield curve is a key step in building a theoretical spot rate yield curve, which then helps price coupon-paying bonds more accurately. You use a method called bootstrapping to get arbitrage-free forward interest rates. Since government Treasury bills don't have data for every period, bootstrapping fills those gaps to construct the yield curve.

Consider bonds with $100 face values and maturities of 0.5, 1, 1.5, and 2 years, with par yields of 2%, 2.3%, 2.6%, and 3% respectively. Coupons pay semi-annually, so the 0.5-year bond has one payment, making its yield equal to the 2% par rate.

For the one-year bond, there are two payments: the first is $100 x (0.023/2) = $1.15 after six months, discounted at the 2% six-month spot rate. The second is $1.15 + $100 = $101.15. To find the discount rate for par value of $100, solve: $100 = $1.15/(1 + 0.02/2) + $101.15/(1 + x/2)^2, which simplifies to x = 2.302%, the one-year spot rate.

You can extend this process to calculate spot rates for the 1.5-year and 2-year bonds similarly.

The Bottom Line

In summary, the par yield curve depicts hypothetical Treasury securities at par, where coupon rates match yields to maturity. It differs from the spot yield curve, which usually sits above it, and assists in setting coupon rates for new bonds to trade at par. Through bootstrapping, you can derive forward rates to bridge data gaps, gaining better insight into interest rates across maturities. Grasping this curve equips you to make smarter choices in bond pricing and investments.