# Par Yield Curve: Definition, Calculation, and Comparison to Spot Curve

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*— Bloomberg/Getty Images*

> **Definition**
>
> A par yield curve is a graphical representation of the yields of hypothetical Treasury securities with prices at par.

## What Is the Par Yield Curve?

The par yield curve graphically represents the yields of [Treasury securities](https://www.investopedia.com/articles/investing/073113/introduction-treasury-securities.asp) priced at [par](https://www.investopedia.com/terms/p/parvalue.asp), where the coupon rate matches the bond's [yield to maturity](https://www.investopedia.com/terms/y/yieldtomaturity.asp). This method of analysis helps investors evaluate the price and yield dynamics of bonds, offering insights distinct from those provided by the [spot](https://www.investopedia.com/terms/s/spot_rate_yield_curve.asp) and forward yield curves.

### Key Takeaways

*   A par yield curve graphically represents the yields of hypothetical Treasury securities priced at par, providing a distinct comparison to spot and forward yield curves.  
    
*   For a bond to trade at par, its coupon rate must equal the yield to maturity, demonstrating the alignment of interest rates with bond pricing.  
    
*   Par yield curves generally fall below both spot and forward yield curves under normal market conditions due to differing maturities and interest rate expectations.  
    
*   Bootstrapping is a method used to derive forward interest rates and complete the par yield curve by filling in gaps left by missing Treasury bill data.  
    
*   The par yield curve helps determine the necessary coupon rate for new bonds to sell at par, facilitating bond pricing decisions.

## Exploring the Dynamics of the Par Yield Curve

The yield curve is a graph that shows the relationship between interest rates and [bond yields](https://www.investopedia.com/articles/investing/022516/understanding-different-types-bond-yields.asp) of various maturities, ranging from three-month [Treasury bills](https://www.investopedia.com/terms/t/treasurybill.asp) to 30-year [Treasury bonds](https://www.investopedia.com/terms/t/treasurybond.asp). The y-axis shows interest rates, while the x-axis represents increasing time durations.

Since short-term bonds typically have lower yields than longer-term bonds, the curve slopes upwards to the right. When people mention the yield curve, they usually mean the spot yield curve for [risk-free bonds](https://www.investopedia.com/terms/r/riskfreeasset.asp). However, there are some instances where another type of yield curve is referred to—the par yield curve.

The par yield curve graphs the YTM of coupon-paying bonds of different maturity dates. The yield to maturity is the return that a bond investor expects to make assuming the bond will be held until maturity. When a bond is issued at [par](https://www.investopedia.com/terms/p/parvalue.asp), its YTM equals the [coupon rate](https://www.investopedia.com/terms/c/coupon-rate.asp). As interest rates change, the YTM adjusts accordingly.

If interest rates drop after a bond is issued, the bond's value goes up because its coupon rate is higher than current rates. In this case, the coupon rate will be higher than the YTM. YTM is the [discount rate](https://www.investopedia.com/terms/d/discountrate.asp) at which a bond's future cash flows equal its current price.

A par yield is the coupon rate at which bond prices are zero. A par yield curve represents bonds that are trading at par. In other words, the par yield curve is a plot of the yield to maturity against [term to maturity](https://www.investopedia.com/terms/t/termtomaturity.asp) for a group of bonds priced at par. It helps find the coupon rate a new bond with a set maturity needs to sell at par today.

The par yield curve gives a yield that is used to discount multiple cash flows for a coupon-paying bond. It uses the spot yield curve data to discount each coupon by the right spot rate. The spot yield curve sits above the par yield curve in upward slopes and below in downward slopes.

![Image](https://www.investopedia.com/thmb/mcStQxr9CcJETIp8RRNH6EEPK64=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/dotdash_Final_Par_Yield_Curve_Apr_2020-01-3d27bef7ca0c4320ae2a5699fb798f47.jpg)

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

Deriving a par yield curve is one step toward creating a theoretical spot rate yield curve, which is then used to more accurately price a coupon-paying bond. A method known as [bootstrapping](https://www.investopedia.com/terms/b/bootstrapping.asp) is used to derive the arbitrage-free [forward interest rates](https://www.investopedia.com/terms/f/forwardrate.asp). Since Treasury bills offered by the government do not have data for every period, the bootstrapping method is used mainly to fill in the missing figures in order to derive the yield curve. For example, consider these bonds with face values of $100 and maturities of six months, one year, 18 months, and two years.

Since coupon payments are made semi-annually, the six-month bond has only one payment. Its yield is, therefore, equal to the par rate, which is 2%. The one-year bond will have two payments made after six months. The first payment will be $100 x (0.023/2) = $1.15. This interest payment should be discounted by 2%, which is the spot rate for six months. The second payment will be the sum of the coupon payment and principal repayment = $1.15 + $100 = $101.15. We need to find the rate at which this payment should be discounted to get a par value of $100. The calculation is:

*   $100 = $1.15/(1 + (0.02/2)) + $101.15/(1 + (x/2)) 
*   $100 = 1.1386 + $101.15/(1 + (x/2))
*   $98.86 = $101.15/(1 + (x/2)) 
*   (1 + (x/2))  = $101.15/$98.86
*   1 + (x/2) = √1.0232
*   x/2 = 1.0115 – 1
*   x = 2.302%

This is the zero-coupon rate for a one-year bond or the one-year spot rate. We can calculate the spot rate for the other bonds maturing in 18 months and two years using this process.

## The Bottom Line

The par yield curve represents hypothetical Treasury securities trading at par, where the coupon rate equals the yield to maturity. Unlike the spot yield curve, which typically lies above the par yield curve, the par curve helps determine appropriate coupon rates for new bonds that sell at par today. By employing methods like bootstrapping, investors can derive forward interest rates that fill in market data gaps, thereby fostering a deeper understanding of interest rate behavior over different maturities. Understanding the nuances of the par yield curve can empower investors to make more informed decisions about bond pricing and investment strategies.