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%


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%


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 And what I'm going to suggest uses a single curve framework which is long outdated.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:

$$1.03\% * (2 D_1 + 2 D_4 ) = 1 - D_4$$

• Thanks a lot for the answer. How would the method look like for multi-curve?
– MCM
Feb 20 '18 at 6:58
• That article uses bootstrapping. Multiple curves are often too complicated to use this analytical method. So they rely on a numerical optimizer: stuff all the variables in, give the curves flexibility at certain points and find the optimal curves that minimise the sum of squares of difference to variable inputs. That article is good for covering a lot of bases. I think the conclusions are not the best since they prioritise things that aren't necessary. Here is a modern trading approach in Darbyshire "Pricing and Trading Interest Rate Derivatives".
– Attack68
Feb 20 '18 at 7:08