Interpolating Bond Yields Through Risk-Free Curves and Credit Spreads
Summary
The document considers how to estimate a bond yield at an intermediate maturity when two other bonds provide reference yields. For risky bullet bonds, its main recommendation is to separate the yield into a risk-free benchmark and a credit or liquidity spread. Use observable benchmark rates at the relevant maturities, interpolate between the two known spreads by time, then combine the estimated spread with the intermediate benchmark rate. Linear interpolation is offered as a straightforward method when spreads are similar.
Large differences between endpoint spreads may signal a material credit or debt repayment feature, so the curve should be investigated and the intermediate spread may need a reasoned adjustment. For amortizing or high-coupon bonds, the answer outlines a more detailed approach: project cash flows, infer survival probabilities or hazard rates from observed bonds and the swap curve, then price the intermediate bond. Other responses mention simple linear interpolation and Nelson–Siegel–Svensson fitting for broader maturity datasets. No market data comparison establishes which method performs best; the appropriate approach depends on bond features and curve context.
Key ideas
- For risky bonds, separate the risk-free benchmark rate from the credit and liquidity spread before interpolating.
- Time-weighted linear interpolation can estimate the intermediate spread when endpoint spreads are close.
- Unusually different spreads call for investigation of credit risk and repayment schedules.
- Amortizing or high-coupon bonds may require cash-flow-level modeling and survival assumptions.
- A fitted curve model can help represent yields across many maturities.
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Full text
# Interpolating a yield from two yields (giving more weight to one of the two)
# Interpolating a yield from two yields (giving more weight to one of the two)
I would like some guidance with the following please.
Suppose I have two yields:
```
2020 bond --> yield = 2.5
2027 bond --> yield = X (to be interpolated)
2030 bond --> yield = 2.8
```
How could I interpolate the 2027 bond yield, considering that it should be closer to the yield from 2030? I am doing this with excel
Thank you.
## Answer by Dimitri Vulis (score 2)
https://quant.stackexchange.com/a/61368
If these are risky (e.g., corporate) bullet bonds, then I would not interpolate the yield directly, because their yield has two distinct components: a risk-free rate and an additional spread (to compensate the bond holder for credit, liquidity, etc risks). You should look for observable risk-free benchmark (probably swap curve, but maybe treasury, depending on the context) for each of the 3 maturities. Then you have no need to interpolate those.
You then have spreads $s_1,s_3$ at maturities $m_1,m_3$. (I like Z-spread, but you may prefer asset-swap spread or some other spread, depending on the context.) Normally, the spreads should be close and so your choice of interpolation is not very material. You can interpolate the spread linearly with respect to the time if you like $w_1 = \frac{m_3-m_2}{m_3-m_1}$, $w_3=1-w_1=\frac{m_2-m_1}{m_3-m_1}$, and $s_2=w_1s1+w_3s_3$.
However if $s_1,s_3$ are very different, then you should dig in deeper (do your due diligence) and undersand why they are. E.g., if $s_3$ is much wider than $s_1$ because a lot of debt (including perhaps the second bond) needs to be repaid between $m_1$ and $m_3$, then you should consider where $m_2$ fits into this and perhaps manually give more weight to the wider spread.
Then you back out the second bond from the second risk-free benchmark and the spread $s_2$.
If any bonds amortize or pay very high coupon, then you may want to consider risk-free benchmarks at additional times for more cash flows than just the maturities. For example, you could project all the cash flows of the 3 bonds; assume some recovery in case of default (e.g. 40%, does not affect the result much); solve for a survival curve using first and third bonds and all of the swap curve (i.e. solve for constant hazard rate $h_1$ from now to second bond's maturity, and for constant hazard rate $h_2$ from second bond's maturity to third bond's maturity); price the second bond using this survival curve.
## Answer by user42108 (score 1)
https://quant.stackexchange.com/a/60699
Linear interpolation. See, for e.g., https://en.wikipedia.org/wiki/Linear_interpolation. Easily implemented in Excel.
## Answer by Whispered (score 1)
https://quant.stackexchange.com/a/60714
If you has a lot of points with wide matuirities you can use the specific Nelson–Siegel–Svensson (NSS) model. It closely follow academic researches for short-, mid- and long-term structures.
Real implementation fot python - https://github.com/luphord/nelson_siegel_svenssonShown 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.