# What Is a Par Yield Curve?

The par yield curve is a method of determining the yield curve for Treasury securities by pricing all maturities at par value. When a bond is priced at par value, the interest rate should be equal to the coupon rate paid on the bond. Under normal circumstances, the par yield will generally be lower than both the spot and forward yield curves.

## Basics

In financial analysis, the par yield curve visually depicts yields for theoretical Treasury securities trading at par. Notably, on this curve, the coupon rate aligns precisely with the yield to maturity (YTM) of the security, resulting in the Treasury bond trading at par value. This graphical representation finds itself in comparison with both the spot yield curve and the forward yield curve concerning Treasuries.

## Exploring Par Yield Curves

The yield curve serves as a visual representation of the correlation between interest rates and bond yields across diverse maturities, spanning from three-month Treasury bills to 30-year Treasury bonds. When displayed on a graph, the y-axis signifies interest rates, while the x-axis indicates increasing time durations.

Typically, short-term bonds yield lower returns compared to their long-term counterparts, resulting in an upward-sloping curve. When discussing the yield curve, it commonly refers to the spot yield curve, particularly for risk-free bonds. Nevertheless, another variant, the par yield curve, is occasionally referenced.

The par yield curve illustrates the Yield to Maturity (YTM) of coupon-paying bonds with varying maturity dates. YTM represents the anticipated return for a bond held until maturity. Bonds issued at par have a YTM identical to the coupon rate. As interest rates fluctuate, YTM adjusts to mirror the current interest rate environment.

For instance, if interest rates decrease post-bond issuance, the bond's value rises because the fixed coupon rate now surpasses the interest rate. Consequently, the YTM becomes lower than the coupon rate. Functionally, YTM acts as the discount rate, aligning future cash flows (coupons and principal) with the bond's current price.

A par yield signifies the coupon rate, rendering bond prices at zero. The corresponding par yield curve charts bond trading at par, displaying the YTM against term to maturity for bonds priced at par. It aids in determining the coupon rate for a new bond to sell at par presently. The par yield curve utilizes spot yield curve data, known as the zero percent coupon curve, to appropriately discount each coupon with the corresponding spot rate.

Notably, when the par yield curve ascends, the duration is lengthier on the spot yield curve, consistently positioning it above the par yield curve. Conversely, in a descending par yield curve scenario, the spot yield curve consistently lies beneath it.

## Constructing the Par Yield Curve Through Bootstrapping

The development of a par yield curve represents a crucial step in formulating a theoretical spot rate yield curve, facilitating more precise pricing of coupon-paying bonds. Employing the bootstrapping method becomes imperative, particularly when government-issued Treasury bills lack data for every interval, necessitating the completion of missing figures to derive the yield curve. Consider the following bonds with face values of $100 and maturities spanning six months, one year, 18 months, and two years.

Maturity (years) | 0.5 | 1 | 1.5 | 2 |

Par yield | 2% | 2.3% | 2.6% | 3% |

Given that coupon payments occur semi-annually, the six-month bond has a singular payment, equating its yield to the par rate of 2%. In contrast, the one-year bond encompasses two payments, with the first being $100 x (0.023/2) = $1.15. This payment is discounted by the six-month spot rate of 2%. The second payment, comprising the coupon and principal, totals $1.15 + $100 = $101.15. To derive the rate discounting this sum to achieve a par value of $100, the calculation unfolds:

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

This outcome denotes the zero-coupon rate for a one-year bond or the one-year spot rate. Subsequent spot rates for bonds maturing in 18 months and two years can be determined using a similar process.

## Conclusion

The par yield curve emerges as a valuable tool for delineating the yield curve of Treasury securities, employing a pricing strategy that aligns all maturities at par value. When bonds are priced at par, the interest rate mirrors the coupon rate, a norm that typically positions the par yield below both spot and forward yield curves. This analytical approach aids in comprehending the intricacies of interest rate movements across various maturities, contributing to a nuanced understanding of bond pricing dynamics. Overall, the par yield curve is a pivotal element in the broader landscape of financial analysis, shedding light on the intricacies of bond valuation and market trends.