# Algebra issue related to Insurance Premium Calc

• Apr 24th 2010, 06:23 AM
Digit100100
Algebra issue related to Insurance Premium Calc
Hi All,

I am stuck with a problem which I feel certain should be solved via algebra, but for the life of me, the soluton evades me. I am interested in calculating a combined insurance premium, where the premium itself is part of the insured amount. I have solved the solution for where there is one cost. I am interested in scaling the formula. The difficulty is probably easier to understand if I put the example into equations:

C = Cost being insured(known)
F = Interest Factor to apply to calculate interest on the premium (known)
R = Premium Rate to apply to calculate the premium (known)
I = Interest on Premium (to be calculated, but also equal to F.P)
P = Premium (to be calculated)

Start point is :
P = (C + P + I)R ie Premium is equal to the cost being insured + the premium itself + interest charged on that premium with the sum being multiplied by the premium rate.

We also know that I = F.P (ie the interest factor multiplied by the premium gives the interest on the premium).

So I rephrased the above as : P = (C + P + F.P)R and solved for P = CR / (1-F.R-R)

That was all simple and my algebra coped easily and the world was good. However, I am trying to solve something that starts as one level more complex than that, and scales up quite dramaticaly. In this example, I have 2 of everything, 2 costs, 2 interest factors, 2 premium rates and the trick is that I have to insure on both sides the full premium and the interest related to both premium parts. So let's present as follows:

C1 and C2 = Costs being insured (known)
F1 and F2 = Interest Factors for each part of the premium (known)
R1 and R2 = Premium Rates for each constituent part (known)
I1 and I2 = the interest cost relating to the payment of each premium part.

My equations are thus the following:
P = P1 + P2 (Total Premium = Premium on C1 element + Premium on C2 element)
I1 = F1P1
I2 = F2P2
P1 = (C1 + P + I1 + I2)R1 (In both instances the total premium and the corresponding interest charges have to be insured)
P2 = (C2 + P + I1 + I2)R2

Now I have presented the P1 and P2 equations as this:
P1 = (C1 + P1 + P2 + F1P1 + F2P2) R1
P2 = (C2 + P1 + P2 + F1P1 + F2P2) R2

I have 2 equations where there are only 2 unknowns (P1 andP2), logic tells me I have to be able to solve this or re-present it to be P1 + P2 (as in total premium) on one side of the equation, but my efforts have gone horribly wrong. Can anyone help?

The next step would be to consider 3 costs etc...but at this stage I am stuck at this step.

Many thanks,

Digit100100

Sorry, I've been tinkering - taking the original equations:
P1 = (C1 + P1 + P2 + F1P1 + F2P2) R1
P2 = (C2 + P1 + P2 + F1P1 + F2P2) R2

You can rephrase the second equation to be P2 = (C2R2 + P1R2 + F1P1R2)/ (1-R2-F2R2) (which starts to look a little like my original simple equation, which hopefully implies I will eventually get to a scalable answer)

Now if you substitute the solution for P2 into the P1 equation and manage the algebra, there must be a solution for P1...I just don't know if I am good enough to do it!