# Bootstrapping The Spot Yield Curve

I have some questions regarding the bootstrapping technique for constructing the spot yield curve. Assume that I want to fit the spot yield curve for U.S Treasury securities and I observe the prices for on-the-run securities (1m, 3m, 6m, 1y, 2y, 3y, 5y, 7y, 10y, 20y, 30y).

Since T-bills (1m, 3m, 6m, 1y) are zero-coupon bonds, I simply calculate the yield to maturity for these bonds and then I have the corresponding spot rates. For example, assume that the price for the 6 months T-bill is given by \$98 and the price for the 1 year T-bill is given by \$95, then I calculate: \begin{align*} r(0,0.5)&=\left(\frac{100}{98}\right)^{1/0.5}-1=4.12\% \\ r(0,1)&=\frac{100}{95}-1=5.26\% \\ \end{align*} (In my notation $$r(0,t)$$ is the spot rate for maturity $$t$$ at time $$t=0$$). To get the spot rate for a maturity between 6 months and 1 year I use some type of interpolation technique. For example, using linear interpolation I get for the 9-months spot yield: \begin{align*} r(0,0.75)=\frac{0.75-0.5}{1-0.5}\cdot5.26\%+\frac{1-0.75}{1-0.5}\cdot 4.12\%=4.69\% \end{align*} Now I want to calculate the 2 year spot rate. Assume that the observed price of the 2 year T-note is \95 and the coupon rate is 5%, then I need to solve the equation \begin{align*} 95=\frac{2.5}{(1+r(0,0.5))^{0.5}}+\frac{2.5}{(1+r(0,1))^{1}}+\frac{2.5}{(1+r(0,1.5))^{1.5}}+\frac{102.5}{(1+r(0,2))^{2}} \end{align*} for $$r(0,2)$$. But I do not observe a 1.5y Treasury security. How can I calculate the 1.5y spot rate ? In this forum I read that the solution is to use some interpolation technique. However, i wonder between which values we interpolate. In the example above we interpolate the value between two known spot rates, but in this case the last observable spot rate is the one for the 1 year T-bill and there is no observable spot rate for maturities longer than 1.5 years. Intuitively, I would calculate $$r(0,1.5)$$ by extrapolation. For example, using a linear extrapolation: \begin{align*} r(0,1.5)=4.12\%+\frac{5.26\%-4.12\%}{1-0.5}\cdot(1.5-0.5)=6.4\% \end{align*} But especially for large gaps like 20y and 30y this technique seems to be not appropriate. Any answers are welcome. • As a practical matter, you can observe in secondary market U.S. treasury bonds that were issued at 30Y, and now have almost all of 21, 22, 23..29 years to maturity. Still, sometimes you need to interpolate. – Dimitri Vulis Apr 6 at 12:14 • Hello Dimitri, thanks for your answer. It's interesting to hear your insights from the field. Assuming that no bond with a maturity of 1.5 years is observable, would the way of calculating the 1.5 spot rate be justifiable? Furthermore, what are your experiences with different interpolation methods? Is a simple linear interpolation sufficient in the regular case or should one make time and effort and try other procedures ? – Jonas_Dim Apr 6 at 14:00 • why don't you setr(0,1.5) = \frac{1}{2} r(0,1) + \frac{1}{2}r(0,2)\$ ? – Antoine Conze Apr 6 at 14:03