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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 $t-1$:

$$\text{min}\;\lambda\cdot\sum_i\left(P_i-\sum_t\frac{F_{it}}{\left(1+r_t\right)^t}\right)^2\\+\left(1-\lambda\right)\sum_t\left(f_{(t)(t+1)}-f_{(t-1)(t)}\right)^2$$

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.

Questions:

  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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