Conceptual Understanding of Rate / Volume Analysis for Balance Sheet changes Not sure if this is the right place to ask this but I searched and didn't find this question already asked. I am having a lot of trouble conceptually understanding the formulas behind a rate / volume analysis for changes to a bank's balance sheet. I know this is just a specific application of a more general question (apportioning change to different factors) but this is the application within which I am working. Below is an example and then my question.


*

*Time Period 1: Balances = 100, Interest Rate = 1%, Income = 100 * 1% = 1

*Time Period 2: Balances = 200, Interest Rate = 2%, Income = 200 * 2% = 4

*Change in Income = 4 - 1 = 3


I am trying to explain how much of the increase in income is due to the balance increase and how much is due to the interest rate increase. The way I was taught to do this, and everything I've read in the last hour or so of googling, is below:


*

*Change due to volumes = (200 - 100) * 1% = 100 * 1% = 1

*Change due to rates = (2% - 1%) * 200 = 1% * 200 = 2


The math here works, and I understand what we're doing conceptually by calculating the change due to volumes (in the absence of any rate increase this is the income that is attributable to our observed volume increase), but I don't understand what we're doing conceptually by calculating the change due to rates; it seems like we should be multiplying the change in rates times the old balances i.e. calculating increased income in the absence of any balance increase. That math doesn't work, though.
Assuming these are the correct calculations for apportioning change (and if they aren't please let me know!), what is the conceptual explanation behind what we're doing to calculate rate change? And is my understanding of change due to volumes incorrect?
 A: You're correct!
Depending on your predilection, you can think of this as a calculus or a geometry problem.
There's a nice graphical presentation of this here. Don't worry about the write-up, just look at the picture. 
https://www.linkedin.com/pulse/variance-analysis-volume-mix-price-fx-rate-umit-coskun/
You're quite correct in your interpretation. You should be multiplying the rate change times the old balance, not the new balance.
The math doesn't work if you do that, though, because there's actually a 3rd term to explain the variance -- a cross-effect. This is the the extra income earned from the incremental rate, on the incremental balance -- in your case, 1% x 100 = 1, which ties our your formulas.
In the formula you've given, you've lumped in this 3rd amount in with the rate change amount. That's maybe simpler, cleaner, and not a big deal, but in the most important sense -- pedantically -- not quite correct. 
The graphical explanation makes this pretty clear - have a look!
Or you can think of this in a calculus sense of a total derivative broken down into partial derivatives. df(ab)/dx = da/dxb + db/dxa + da/dxdb/dx (I think I got that right). 
A: I think it would be easier if you imagine an interim state where
Balance  = 200, and interest rate = 1%.
The change can be regarded as two small changes.


*

*From Period 1 to interim state. Balance increased from 100 to 200. But rate stays the same. 

*From interim state to Period 2. Balance stays as 200. But rate increased from 1% to 2%.

