STRIPS Bonds and the Term Structure of Interest Rates

Learning Objectives

• We’ll introduce the bootstrapping method to estimate the term structure of interest rates. The bootstrapping method is a technique for constructing a zero-coupon yield curve from the prices of zero-coupon bonds with different maturities created by stripping a single coupon bond.

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Introdcution

Registered Interest and Principal of Securities (STRIPS) bonds are a unique type of fixed-income investment that provides an alternative way to access the income and coupon payments of Treasury securities. STRIPS bonds are created by separating a Treasury security’s coupon and principal components and trading them as individual zero-coupon securities (Fig. 11).

Fig. 11 Schematic of United States Treasury Registered Interest and Principal of Securities (STRIPS) debt instrument.

For example, the 5-year Treasury note with annual coupon payments of \(C\) USD and a face (par) value of \(V_{P}\) (USD) in Fig. 11 can be stripped into six separate zero-coupon securities, i.e., five zero-coupon bonds, each with face values of \(C\) and maturity of T = 1,2,3,4 and 5 years, and a six security with face (par) value of \(V_{P}\) USD with a duration of \(T\) = 5 years. In the general case, a treasury note or bond with \(N=\lambda{T}\) coupon payments, where \(T\) denotes the maturity in years, and \(\lambda\) represents the number of coupon payments per year, can be stripped into \(N+1\) separate zero-coupon securities.

Term Structure of Interest Rates and STRIPS

Beyond thier immediate value as investment tools, STRIPS bonds are interesting because they provide another look at the Term Structure of Interest Rates, i.e., how the change in the short rates influences the price and yeild the bond. The process of using the prices of STRIPS bonds to estimate the term structure of interest rates is known as bootstrapping.

Bootstrapping

We can estimate the short rates, which represent the market rate of interest between periods \(j\rightarrow{j+1}\) and are denoted by \(r_{j+1,j}\), by analyzing the prices of the various STRIPS zero coupon products based on their maturity. Using a discrete discounting model, the short rates are calculated according.:

\[\begin{split} \begin{eqnarray} \frac{V_{P,1}}{V_{B,1}} & = & \left(1+r_{1,0}\right) \\ \frac{V_{P,2}}{V_{B,2}} & = & \left(1+r_{2,1}\right)\cdot\left(1+r_{1,0}\right) \\ \vdots & = & \vdots \\ \frac{V_{P,N}}{V_{B,N}} & = & \prod_{i=1}^{N}\left(1+r_{i,i-1}\right) \\ \end{eqnarray} \end{split}\]

where \(V_{P,i}\) and \(V_{B,i}\) denote the face (par) value and price of the \(i^{th}\) zero-coupon bond (both of which are known). Thus, we can solve for \(r_{1,0}\), then insert that into the following expression to solve for \(r_{2,1}\), and so on. Systematically, we can solve for the log-transformed short rates as a system of linear algebraic equations (LAEs) of the from:

\[ \mathbf{A}\mathbf{x} = \mathbf{b} \]

where \(x_{i} = \log\left(1+r_{i,i-1}\right)\), \(b_{i} = \log\left(V_{P,i}/V_{B,i}\right)\) and \(\mathbf{A}\) is a lower-triangular matrix of 1’s. We solve for the log-transformed short rates by computing the inverse of the matrix \(\mathbf{A}\):

\[ \mathbf{x} = \mathbf{A}^{-1}\mathbf{b} \]

and then transform these back to linear coordinates for each period:

\[ r_{i,i-1} = 10^{x_{i}} - 1 \]

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Summary

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