Interpolating Zero Rates and the Limits of Sparse Yield-Curve Data
Summary
The document asks how to build a zero or spot curve from two observations: one-year and three-year zero-coupon swap rates. It explains that interpolation is underdetermined because the observed rates do not reveal the path of intermediate forward rates. Two possible annual forward-rate patterns illustrate how different curves can fit the same endpoints, and either pattern could be used to derive overnight rates across the period.
The answer notes that linear interpolation is a practical approximation, not a uniquely justified solution. A comparison of interpolated Treasury yields with an exponential yield fit across maturities shows close agreement in the cited example. This suggests smooth yield changes can make simple interpolation effective locally. The evidence is limited to that example, however, and sparse market inputs cannot identify the true curve or daily overnight rates without additional assumptions or data.
Key ideas
- Two zero-rate observations do not uniquely determine the intermediate forward-rate path.
- Different forward-rate patterns can match the same observed curve endpoints.
- Linear interpolation is a practical approximation rather than a uniquely implied method.
- A cited Treasury example finds linear interpolation close to an exponential fit.
- Daily overnight rates require more information or assumptions than the two quoted rates provide.
Tags
Full text
# Yield curve interpolation
# Yield curve interpolation
I'm trying to build a zero/spot curve and have two pieces of information.
- 1yr zero coupon swap = 1%
- 3yr zero coupon swap = 3%
My initial guess was to linearly interpolate which produces a linear curve with slope 1. However, this seems very hand wavy (not rigorously justified) and I'm not convinced it is right. Is there a better approach to tackle this?
My end goal is to find overnight rates for each day in the 3yr period.
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/36275
Consider the possible forward rates in each of the three years. I.e. In X for 1 year forward rates, where X is 0,1 or 2. Possible solutions (ignoring compounding for simplicity) include the following :
(1,3,5): Forward rates are a linear function of time. And (1,4,4): Forward rates shoot up towards 4 and then stay constant.
These are very different solutions but both are possible. Each solution can be used to produce overnight rates- for example the overnight forward rate for the first solution is 2x, where X is the forward time.
However, an intermediate solution seems more likely. Forward rates are not usually linear, nor do they reach some asymptote within a year. You don't have enough information to determine this more accurately.
## Answer by caverac (score 0)
https://quant.stackexchange.com/a/36231
If you do not have other information about the underlying, I am afraid this is pretty much all you can do. However, it is not that bad as you think. To give you an idea below is the treasury yield rates at three different dates: today (09/28/2017), one year (09/28/2016) and two years (09/28/2016) ago
For each curve I fit a model (dashed lines) of the form
$$ {\rm yield} = ae^{-bt} + c $$
And these are the estimates for 09/28/2017
$$ \begin{array}{c|rrr} t~({\rm yr}) & \text{exp. fit} & \text{interp.} & \text{actual} \\ \hline 1.0 & 1.24 & 1.31 & 1.31 \\ 1.5 & 1.34 & 1.38 & - \\ 2.0 & 1.43 & 1.45 & 1.45 \\ 2.5 & 1.52 & 1.52 & - \\ 3.0 & 1.60 & 1.59 & 1.59 \\ \end{array} $$
which shows that the linear interpolation is actually really close to the exponential fit. The reason behind this is that the time variation of the yields is smooth enough to be approximated by a low order expansion around a given point in timeShown 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.