Yield Curve Smoothing by Penalizing Forward-Rate Slope
Summary
The document describes a government-bond curve fit that minimizes squared bond-price errors while imposing a monotonicity condition on discount factors. This price-focused fit can produce a pronounced hump. It then presents a regularized objective combining price errors with a penalty on changes between adjacent forward rates. The parameter lambda controls the tradeoff: lowering it gives more weight to forward-curve smoothness and less to fitting observed bond prices, which can reduce or remove the hump.
The text frames questions about why smoother forwards might be financially useful, including for asset-liability matching, and what justifies accepting larger spot-rate estimation errors. It does not provide answers, empirical comparisons, a criterion for selecting lambda, or evidence that a smoother curve improves hedging or liability valuation. The method therefore highlights a calibration tradeoff rather than establishing that smoothness is always desirable; the appropriate balance depends on the intended use and the quality of the bond data.
Key ideas
- A price-error objective can produce a yield curve with a pronounced hump.
- A penalty on adjacent forward-rate changes regularizes the fitted curve.
- The smoothing weight trades bond-price fit against forward-curve smoothness.
- Increasing smoothness can raise the discrepancy between estimated and observed rates.
- The document poses, but does not resolve, when this tradeoff benefits asset-liability matching.
Tags
Full text
# Smoothening yield curve by minimizing forward curve slope
# Smoothening yield curve by minimizing forward curve slope
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:
- Why does minimizing the forward curve slope smoothen the yield curve? What is happening financially?
- Why is a smooth yield curve desirable in the context of asset liability matching?
- 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.