# Calculating profit margin

+1 vote
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edited May 13, 2018

Pls can someone help me with this question.

Assume and interest rate of 12% pa. A 5 year business insurance policy has monthly-premiums of 194.26. The pricing basis allows for 0.025 claims a year, occurring uniformly over the year. The average value of the claim in year 1 is 50000, but is expected to grow by 7% p.a. The policy has regular costs of R500 per year, incurred in advance. What is the profit margin used to price this policy?

answered May 14, 2018 by (2,850 points)
selected May 24, 2018 by Lwazi

To start off, remember the formula for calculating office premiums: $$EPV(OP)=EPV(Claims)+EPV(Expenses)+EPV(Profit\ Margin)+EPV(Risk\ Margin)$$

First, since the interest rate is an effective annual one, and premiums are monthly, we must convert the interest rate:

$$j=\left(1+i \right)^{1/12}-1=0.9488792935\%$$

Now, consider the parts of the first formula one by one:

$$EPV(OP)=194.26 ä_{\overline{5\times12|}@j}=194.26 \times \frac{1-V^{60}_{j}}{1-V_{j}}=R8939.916658$$

$$EPV(Exp)=500ä_{\overline{5|}@i}=R2018.674673$$

Now, for claims it is easiest to draw out a timeline and then write out the $$EPV$$ in full. We can then simplify it into something that's easier to calculate. Doing this (note the uniform distribution assumption), we get:

$$EPV(Claims)=0.025\times 50000 \times V^{0.5}_{i} +0.025 \times 50000 \times (1+7\%) \times V^{1.5}_{i} + 0.025 \times 50000 \times (1+7\%)^2 \times V^{2.5}_{i} + ... +0.025 \times 50000 \times (1+7\%)^4 \times V^{4.5}_{i}$$

Now we can tackle this as we would any increasing annuity: by taking out common factors!

$$EPV(Claims)=0.025 \times 50000 \times V^{0.5}_{i} \times \left(1+ (1+7\%)V^{1}_{i}+ (1+7\%)^{2}V^{2}_{i} +...+ (1+7\%)^{4}V^{4}_{i} \right)$$

Now we need to work out our new interest rate, taking the increase into account. We'll call it $$k$$.

$$k=\frac{1+12\%}{1+7\%}-1=4.672897196\%$$

Take care to note that this is a p.a. rate, since we used $$i$$ in calculating it, and $$i$$ is also a p.a. rate.

Now we can use $$k$$ in calculating $$EPV(Claims)$$:

$$EPV(Claims)=0.025 \times 50000 \times V^{0.5}_{i} \times ä_{\overline{5|}@k} = R5401.41987$$

Finally, plugging our answers (NOT ROUNDED OFF) into the very first equation, we get that

$$EPV(Profit\ Margin)=R1519.82$$

You could of course then go on the calculate what the profit margin is as a percentage of the office premiums but I'll leave that for you to do :)