# What Is the Spot Rate Treasury Curve?

The spot rate Treasury curve is a yield curve built using Treasury spot rates instead of yields. The curve is constructed by connecting the actual spot rates for zero-coupon Treasury bonds. To value a bond accurately, it's good practice to match and discount each coupon payment with the corresponding point on the Treasury spot rate curve. Think of a coupon bond as a collection of zero-coupon bonds, where each coupon represents a small zero-coupon bond that matures when the bondholder receives the coupon.

## Basics

Constructing a benchmark for bond pricing, the spot rate Treasury curve diverges from traditional yield curves by utilizing Treasury spot rates instead of yields. This curve can be derived from on-the-run treasuries, off-the-run treasuries, or a mix of both. Alternatively, a straightforward approach involves employing zero-coupon Treasury bonds' yields, which mirror the spot rate for such bonds.

## Spot Rate Treasury Curve: A Deeper Insight

In the valuation realm, bonds stray from conventional pricing anchored in Treasury yields, opting instead for the dynamic representation of market sentiment through Treasury spot rates. The resultant graphical depiction, known as the spot rate Treasury curve, crystallizes these expectations. Spot rates, encapsulating immediate bond settlements, integrate forecasts of market shifts. Theoretically, the spot rate or yield aligns with a zero-coupon bond sharing the same maturity.

This curve unfurls the yield to maturity (YTM) for a zero-coupon bond, which is pivotal in discounting the maturity cash flow. Employing an iterative or bootstrapping approach, the YTM becomes the linchpin in discounting each coupon payment at the corresponding spot rate for its maturity.

Diverging from a one-size-fits-all approach, bonds with multiple coupon payments throughout their lifespan necessitate a nuanced valuation. Meticulously aligning each coupon payment with the corresponding point on the Treasury spot rate curve ensures accurate pricing of the present value of each coupon.

A conceptual lens reveals a coupon bond as an amalgamation of mini zero-coupon bonds, each maturing when a coupon is disbursed. Precision in valuation demands matching each coupon payment with the corresponding spot rate for a zero-coupon Treasury bond maturing at the same time.

Navigating the extensive Treasury bond market, actual data points elude us across temporal spectrums. Connecting the dots, spot rates for zero-coupon Treasury bonds converge to craft the spot rate Treasury curve, a pivotal tool in discounting intricate coupon payments.

## Illustration of the Spot Rate Treasury Curve

Consider a scenario where a $100 par value, two-year 10% coupon bond is priced using Treasury spot rates. The successive spot rates for the ensuing four periods (each year divided into two) stand at 8%, 8.05%, 8.1%, and 8.12%. Corresponding cash flows amount to $5 (computed as 10% / 2 x $100), $5, $5, and $105 (coupon payment plus maturity principal).

Mapping spot rates against maturities yields the spot rate or zero curve. Employing the bootstrap method, periods are designated as 0.5, 1, 1.5, and 2, representing successive 6-month intervals.

The present value for each respective cash flow is calculated using a prescribed formula:

Theoretically, the bond's market value should be $103.71. However, the actual selling price may differ, as the spot rates used for pricing stem from risk-free Treasuries. To account for increased risk in corporate bonds versus Treasuries, further discounting is necessary. Crucially, the spot rate Treasury curve isn't an accurate gauge of average market yields, given the prevalence of non-zero-coupon bonds.

## Conclusion

The spot rate Treasury curve, based on Treasury spot rates, serves as a dynamic benchmark for precise bond pricing. By connecting actual spot rates for zero-coupon Treasury bonds it ensures accurate valuation. The concept of viewing a coupon bond as a mini zero-coupon bond highlights the need for precision in aligning each coupon payment with the corresponding spot rate. It's crucial to note that the spot rate Treasury curve doesn't perfectly reflect average market yields, given the prevalence of non-zero-coupon bonds.