I run an insurance operation and I am trying to measure the effect of
a change in any variable used in determing premium from one year to
the next.

Example:

Year 1: Premium1 = Revenues1 X Factor11 X Factor12 X . . . X Factor1N

Year 2: Premium2 = Revenues2 X Factor21 X Factor22 X . . . X Factor2N

I want to be able to determine how much the premium changes due to
changes in revenues and factor 1 to N such that their sum = the change
in premium. Has to do with partial derivatives by my 30 year old
calculus training is too rusty to be able to derive the answer.

---

Clarification of Question by baybig-ga on 03 Dec 2005 05:46 PST

Revenues is policyholder revenues in each year as we are writing a
professional liability line

baybig-ga

  As you did not expire the question, I assume issue is still unresolved.

  So, let's look at the simple example

   Y =A * B *C ,

where factors A , B , C depend on some variable or variables.

case one: A, B, C depend on one variable u (e.g. time)
          You do not need partial derivatives, just a product rule:

   Y' = A' *B *C + A * B' *C  + A * B *C'

SEARCH TERM: product rule 
http://archives.math.utk.edu/visual.calculus/2/product_rule.5/

case 2:  A, B, C ...  depend on two (or more) variables u, v, ..

         Then change of Y with u  is measured by a partial dY/du
         and  change of Y with v is measured by a partial  dY/dv
         ...

and each partial derivative is calculated by the same 'product rule' as in
the case of the single variable.

  Then, total change of Y = Delta Y can be apropximated by

                Delta Y = dY/du * delta u  + Dy/dv * delta v  
etc

 http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node88.html


Does that all make sense?

Hedgie

Part of your answer is going to depend on how the book of business you
write changes from the first year to the second.  There is a tendency
for those insureds whose rates went up the most, to non-renew to a
greater extent than other insureds.  So you could wind up with a
somewhat different book of business.

It would help me if you could clarify what you mean by "revenues".  I
think of premium as being equal to the number of insureds times the
average rate.

Revenues is policyholder revenues in each year as we are writing a
professional liability line

The change in premium is approximately (first order approximation) the
change in revenue times the partial derivative wkth respect to revenue
which is f1*...fn + the change in f1* the partial derivative with
respect to f1 which is revenue*f2*...fn + etc. So the answer you want
is

delta p is approximately (delta revenue)*f1*...*fn + (delta
f1)*revenue*f2*...*fn+...+(delta fn)*revenue*f1*...*fn-1.

I think this is what you want.