I am using government bullet bond data and have bootstrapped a yield curve by solving the following optimization which minimizes unweighted price error:

$$\text{min}\sum_i\left(P_i-\sum_t\frac{F_{it}}{\left(1+r_t\right)^t}\right)^2$$ where $i$ denotes a specific bond, and $t$ a specific time. The optimization is subject to the condition: $$\left(1+r_t\right)^t\geq\left(1+r_{t-1}\right)^{t-1} \quad\forall{t}$$

The resulting yield curve displays a pretty significant hump. Now I am told to smoothen it, for which purpose my text introduces a weighting variable $\lambda=\{0.1,0.2,\dots,1\}$ and tags on a term containing the difference between the forward rate at $t$ vs. the forward rate at $t-1$:


Under the condition: $$f_{t\tau}= \begin{cases} r_t, & \text{if $\tau=0$}\\ \left(\frac{\left(1+r_\tau\right)^\tau}{\left(1+r_t\right)^t}\right)^{1/\left(\tau-t\right)}, & \text{if $\tau\gt 0$} \end{cases}\qquad\tau\geq t$$

I need to decide which value of $\lambda$ would be optimal, but I do not fully understand the added optimization term, nor the properties of the yield curve itself. I am hopeful that you can help me make sense of it.

The original term works by minimizing the unweighted price errors, but this leads to bonds with a longer maturity being weighted more during the regression. The new term works by minimizing the slope of the forward curve. As $\lambda$ decreases in value, the weight of the optimization shifts from minimizing the unweighted price errors to minimizing the slope of the forward curve. This leads to a lessening of aforementioned hump displayed by yield curve, and eventually its complete disappearance.


  1. Why does minimizing the forward curve slope smoothen the yield curve? What is happening financially?
  2. Why is a smooth yield curve desirable in the context of asset liability matching?
  3. The variance between estimated and observed spot rate increases as $\lambda$ decreases, i.e. the model becomes more uncertain as minimizing price errors becomes less important - this is not surprising. But what makes curve smoothness worth the cost of increased variance?

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