Determine the percent contribution of each factor to the total change

Question

I apologize for the question title, but I'm not very sure of how to describe what I'm looking for.

Let's say I have three variables:

$a_0 = 1.1$; $b_0 = 0.9$; $c_0 = 1.25$

And lets say that: $T_0 = a_0 * b_0 * c_0 = 1.2375$

Now, lets say that the three variables change as such:

$a_1 = 1.22$; $b_1 = 0.7$; $c_1 = 1.3$

And $T_1 = a_1 * b_1 * c_1 = 1.1102$

How can I determine how much the change in each of the variables (a,b,c) affected the change in T?

Thanks,

Edit: As was suggested in the comments, I'm going to elaborate a bit more as to why I need to know this.

In insurance rate-making, we typically start with a base rate and multiply that base rate by different factors (age, location, etc.) to determine the premium to be charged.

For example, the current premium is:

$a_0 = 1000$

$b_0 = 1.05$

$c_0 = 0.94$

$ premium = 987$

And let's say the proposed premium is this:

$a_1 = 1050$

$b_1 = 0.95$

$c_1 = 1.01$

$premium = 1007.48$

The change in premium is 20.48

I want to know how much of that 20.48 change can be attributed to each of the factors (a,b,c).

I hope that helps clear things up a little.

Answer 1

I'm not sure that you can isolate the contributions of each variable in a simple way such that the percentages will add up to 100. Because you are multiplying them all together, the way they vary will interact with one another. So, what follows is just an ad-hoc attempt at a solution...

One way you could measure how much each change affected the total change is to compare what the result would have been had one of your variables not changed at all: $$ \Delta_a(T)=\frac{a_1\cdot b_1\cdot c_1 - a_0\cdot b_1\cdot c_1}{a_1\cdot b_1\cdot c_1} $$ $$ \Delta_b(T)=\frac{a_1\cdot b_1\cdot c_1 - a_1\cdot b_0\cdot c_1}{a_1\cdot b_1\cdot c_1} $$ $$ \Delta_c(T)=\frac{a_1\cdot b_1\cdot c_1 - a_1\cdot b_1\cdot c_0}{a_1\cdot b_1\cdot c_1} $$As previously mentioned, however, the sum of these three quantities can exceed $1$, which might be undesirable. You can then of course just normalize these quantities by dividing by the sum of all three: $$ \Delta^n_a(T)=\frac{\Delta_a(T)}{\Delta_a(T)+\Delta_b(T)+\Delta_c(T)} $$ $$ \Delta^n_b(T)=\frac{\Delta_b(T)}{\Delta_a(T)+\Delta_b(T)+\Delta_c(T)} $$ $$ \Delta^n_c(T)=\frac{\Delta_c(T)}{\Delta_a(T)+\Delta_b(T)+\Delta_c(T)} $$

EDIT: Right after I posted this, I didn't like it. I will leave it up, but the normalization idea is a bad one because if nothing changes then I have an indeterminate form.