Interpolating Swap Discount Factors with a Linear Assumption
Summary
The document asks how to infer intermediate discount factors from sparse swap quotes, particularly when only the one-year and four-year rates are known. Its answer distinguishes between assuming swap rates interpolate linearly and assuming discount factors themselves are linear. Under the latter assumption, the two intermediate discount factors are placed at evenly spaced points between the endpoint factors.
The answer substitutes that linear discount-factor construction into a simplified par-swap relation to connect the four-year quote to the endpoint factors. It cautions that the relation is simplistic: actual swap conventions may involve different floating payment frequencies, and the cited single-curve framework is outdated. The discussion therefore illustrates how an interpolation assumption closes an under-specified problem, rather than establishing a generally preferred curve-building method. It does not provide a full multi-curve bootstrap or market convention details.
Key ideas
- Sparse swap quotes do not uniquely determine intermediate discount factors without a modeling assumption.
- Linear interpolation of discount factors places intermediate values evenly between the known endpoint factors.
- The answer uses a simplified par-swap equation to relate the four-year rate to those factors.
- The formula omits practical conventions such as floating-leg payment frequency.
- The response warns that its single-curve framework is outdated and should be treated as illustrative.
Tags
Full text
# Linear interpolation Discount factors
# Linear interpolation Discount factors
I am not sure how to perform a linear interpolation between discount fators for swap quotes. Lets say I have the following market quotes:
> `12M 0.670% 2Y 0.630% 3Y 4Y 1.030% `
```
12M 0.670%
2Y 0.630%
3Y
4Y 1.030%
```
Here it is clear how the interpoliation would work as defined below in the equoation:
What is not clear is how I would interpolate between the discount factors if I would have only market quotes of 12M and 4Y as shown below to receive the 2Y and 3Y discount factor:
> `12M 0.670% 2Y 3Y 4Y 1.030% `
```
12M 0.670%
2Y
3Y
4Y 1.030%
```
## Answer by Attack68 (score 1, accepted)
https://quant.stackexchange.com/a/38341
I don't recommend linear interpolation of DFs and the swap rates you are applying this to are either against 12M libor which is illiquid or you are not accounting for Quarterly or Semi-Annual floating sides. And what I'm going to suggest uses a single curve framework which is long outdated. But that being said and given the nature of what's been asked...
You have adopted the simplistic formula: $R_{tenor} = \frac{1-D_{tenor}}{\sum_{i}D_{i}}$ You need to make an assumption about your model, since otherwise it is under parametrised. Lets say that you assume the rates are linearly interpolated then the problem is probably trivial to determine the DFs by bootstrapping, after you calculate the 2Y and 3Y rate.
If instead, you want to have linear DFs between 1Y and 4Y then you have the following: $$D_1 = D_1, \; D_2=D_1 + 1/3 (D_4-D_1), \; D_3 = D_1 + 2/3(D_4-D_1), \; D_4=D_4$$
Inserting that into the equation for the 4Y rate gives:
$$ 1.03\% * (2 D_1 + 2 D_4 ) = 1 - D_4 $$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.