## What Is the Spot Rate Treasury Curve?

The spot rate Treasury curve is a yield curve constructed using Treasury spot rates rather than yields. The spot rate Treasury curve is a useful benchmark for pricing bonds. This type of rate curve can be built from on-the-run treasuries, off-the-run treasuries, or a combination of both. However, the easiest method is to use the yields of zero-coupon Treasury bonds. Calculating the yield of a zero-coupon bond is relatively straightforward, and it is identical to the spot rate for zero-coupon bonds.

### KEY TAKEAWAYS

• The spot rate Treasury curve is a yield curve constructed using Treasury spot rates rather than yields.
• The actual spot rates for zero-coupon Treasury bonds are connected to form the spot rate Treasury curve.
• In order to value a bond properly, it is good practice to match up and discount each coupon payment with the corresponding point on the Treasury spot rate curve.
• A coupon bond can be thought of as a collection of zero-coupon bonds, where each coupon is a small zero-coupon bond that matures when the bondholder receives the coupon.

## Understanding the Spot Rate Treasury Curve

Bonds may be priced based on Treasury spot rates rather than Treasury yields to reflect market expectations of changing interest rates. When spot rates are derived and plotted on a graph, the resulting curve is the spot rate Treasury curve.

Spot rates are prices quoted for immediate bond settlements, so pricing based on spot rates takes into account anticipated changes to market conditions. Theoretically, the spot rate or yield for a particular term to maturity is the same as the yield on a zero-coupon bond with the same maturity.

The spot rate Treasury curve gives the yield to maturity (YTM) for a zero-coupon bond that is used to discount a cash flow at maturity. An iterative or bootstrapping method is used to determine the price of a coupon-paying bond. The YTM is used to discount the first coupon payment at the spot rate for its maturity. The second coupon payment is then discounted at the spot rate for its maturity, and so on.

Bonds typically have multiple coupon payments at different points during the life of the bond. So, it is not theoretically correct to use just one interest rate to discount all of the cash flows. In order to value a bond properly, it is good practice to match up and discount each coupon payment with the corresponding point on the Treasury spot rate curve. This allows us to price the present value of each coupon.

A coupon bond can be thought of as a collection of zero-coupon bonds, where each coupon is a small zero-coupon bond that matures when the bondholder receives the coupon. The correct spot rate for a Treasury bond coupon is the spot rate for a zero-coupon Treasury bond that matures at the same time that a coupon is received. Although the Treasury bond market is vast, real data is not available for all points in time. The actual spot rates for zero-coupon Treasury bonds are connected to form the spot rate Treasury curve. The spot rate Treasury curve can then be used to discount coupon payments.

A coupon bond can be thought of as a collection of zero-coupon bonds, where each coupon is a small zero-coupon bond that matures when the bondholder receives the coupon.

## Example of the Spot Rate Treasury Curve

For example, suppose that a two-year 10% coupon bond with a par value of $100 is being priced using Treasury spot rates. The Treasury spot rates for the subsequent four periods (each year is composed of two periods) are 8%, 8.05%, 8.1%, and 8.12%. The four corresponding cash flows are$5 (calculated as 10% / 2 x $100),$5, $5,$105 (coupon payment plus principal value at maturity). When we plot the spot rates against the maturities, we get the spot rate or the zero curve.

Using the bootstrap method, the number of periods will be designated as 0.5, 1, 1.5, and 2, where 0.5 is the first 6-month period, 1 is the cumulative second 6-month period, and so on.

The present value for each respective cash flow will be:

﻿\begin{aligned} &=\5/1.08^{0.5}+\5/1.0805^1+\5/1.081^{1.5}+\105/\1.0812^2\\ &=\4.81+\4.63+\4.45+\89.82\\ &=\103.71\\ \end{aligned}﻿

Theoretically, the bond should be \$103.71 in the markets. However, this is not necessarily the price at which the bond will ultimately sell. The spot rates used to price bonds reflect rates that are from default-free Treasuries. So, the corporate bond's price will need to be further discounted to account for its increased risk compared to Treasury bonds.

It is important to note that the spot rate Treasury curve is not an accurate indicator of average market yields because most bonds are not zero-coupon.