I'm not 100% sure without knowing the book you use or its conventions, but typically when you match something financially, it means you provide that specific amount.
Now if this is in addition to the interest already generated (and I am making the assumption that it is), then it means that whenever the matched portion is added, you simply add 0.04*A(0) where A(0) is the value (in dollars) of the asset at time t = 0 at the time whenever it is matched.
If it is matched initially then you simply have a formula for the interest of the non-compounded asset model (call it A(t)) and you create a new asset model A*(t) where A*(t) = A(t) + 0.04*A(0). If it's added at another time, then you add it at that time.
I'm thinking that it will be added initially as soon as you initiate the lending agreement so in this case you will compare A*(t) to your other asset model instead of comparing A(t) to the other asset model directly.