Constructing the Term Structure
There are two steps of the construction: 1. bootstrpping; 2. interpolation.
Bootstrapping
Bootstrapping refers to a self-starting process that is supposed to proceed without external inputs.
Assume we have the following hypothetical market information
The key equation is
We can infer 3-month, 6-month, and 1-year zero rates straightforwardly
For the coupon bonds, we first eliminate coupons’ value from the price the obtain “PV” and then use Par value plus last coupon as the “FV” to calculate the .
To calculate forward rate, we have formula
thus
We can derive the forward rates using spot rates.
Interpolation
Interpolation is a statistical method by which known values are used to estimate unkown values. Fitting the yield curve means estimating the yields for unobserved maturities. Interpolations include Linear interpolation and Non-linear methods.
Non-linear methods:
- Nelson-Siegel and its variant Nelson-Seigel-Svensson are leading parametric models
- Splines-based models
The Nelson & Siegel Model
Formula:
where denotes the spot rate; and are the constant parameters; and T is the time to maturity in annual units.
Steps
Step 1: Calibrating the Nelson-Siegel model
Step 2: Applying the calibrated Nelson-Siegel model
Polynomial Splines
Spline originally meant a thin wood or metal slat in East Anglian dialect. By 1985, it had come to mean a flexible ruler used to draw curves. In mathematics, a spline is a numeric function that is pieceswise-defined by polynomial functions.
We are fitting bond price here
Understanding the Yield Curve
Spot rate is the discount rate of a single future cash flow such as a zero-coupon bond:
Forward rate is the interest rate for a loan between two future dates contracted today：
Generally, we have
One-year forward rates versus implied spot rates one year forward
Implied spot rates one year forward, if realized, would make all government bonds earn the same holding period return over the next year
Pure Expectations Hypothesis (PEH)
The hypothesis tells that any yield differences across bonds must imply future rate changes in expectation — implied spot rate one year forward is the result of aggregate expectations. Implications:
- all government bonds have the same near-term expected return regardless of their maturities.
- expected capital gains or losses offset the initial yield differences.
Accoding to Risk Premium Hypothesis (RPH), the forward rate reflects a bond’s expected return (due to embedded risks), implications:
- there are no expected changes in the spot rates
- An n-year bond will receive a one-year return of called rolling yield
and the yield is
(if, in theory, the term structure do not change)
The Impact of Convexity
Convexity differentials across bonds tend to make the Treasury yield curve inverted or “humped”.
where
- Rolling Yield =
- Short-Term Rate =
- Bond Risk Premium =
- Convexity Bias = -0.5convexity
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