Solving for Missing Spot Rates with Linear Interpolation

Summary

This example explains how to infer unobserved spot rates from coupon-bond prices when market inputs exist only at selected maturities. The setup assumes continuously compounded spot rates, known rates at earlier coupon dates, and a known price for a semiannual coupon bond maturing at two years. A linear interpolation assumption connects the one-year and two-year spot rates, thereby expressing the intermediate rate in terms of the unknown terminal rate.

Substituting that relationship into the bond-pricing equation leaves one nonlinear equation for the two-year spot rate. The response recommends an iterative numerical method, such as Newton–Raphson, and the intermediate spot rate then follows from the interpolation rule. This illustrates the method but does not provide the full iteration, convergence checks, or safeguards for unusual inputs. Its result also depends on the assumed interpolation and compounding convention; other curve-fitting assumptions can produce different rates.

Key ideas

• Known coupon-bond prices can constrain spot rates at maturities that lack direct observations.
• Linear interpolation expresses the intermediate spot rate as a function of the unknown longer rate.
• Substituting the interpolated rate into the bond price gives a nonlinear equation for the terminal spot rate.
• An iterative root-finding method such as Newton–Raphson can solve the resulting equation.
• The inferred curve depends on the interpolation and compounding assumptions.

Tags

# Interpolating spot rates given intermittent coupon-bond prices.


# Interpolating spot rates given intermittent coupon-bond prices.












I'm trying to bootstrap spot rates given coupon-paying bond data. To simplify my problem, assume we are working with only 3 given data, the price/coupon rate on semi-annual bonds maturing in 0.5, 1, and 2 years.

I can infer the 0.5 and 1 year spot rates from the given data. How do I infer the 1.5 and 2 year spot rates? If it simplifies the problem, assume that the spot rate changes linearly from 1 to 2 years.

Is there an analytic solution to find the 1.5 and 2 year spot rates? Do I need some iterative process?

Any leads on this problem would help. Thanks!

## Answer by crunch (score 1, accepted)

https://quant.stackexchange.com/a/10284

Assume we have $r(t)$ continuously compounded spot rate for maturity $t$. The price of the 2-year bond with semi-annual coupon $C$ is known to be $P$. We already have $r(0.5)$ and $r(1)$. We need $r(2)$ and $r(1.5) = f(r(1), r(2))$. Then

$$ P = C [e^{-0.5 \times r(0.5)} + e^{-r(1)}+e^{-1.5 \times r(1.5)}] + (1+C)e^{-2 \times r(2)} $$

Using linear interpolation, $r(1.5) = 0.5 [r(2) + r(1)]$. Substituting in, we get:

$$ P = C [e^{-0.5 \times r(0.5)} + e^{-r(1)}+e^{-1.5 \times 0.5 [r(2) + r(1)]}] + (1+C)e^{-2 \times r(2)} $$

The best way to solve for $r(2)$ is with some optimisation technique. If you want to use an iterative approach, a simple Newton-Rhapson will be good enough.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.