# 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%


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: $$D_1 = D_1, \; D_2=D_1 + 1/3 (D_4-D_1), \; D_3 = D_1 + 2/3(D_4-D_1), \; D_4=D_4$$
$$1.03\% * (2 D_1 + 2 D_4 ) = 1 - D_4$$