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Calculating profit margin

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asked May 13 in BUS 1003H - Introduction to Financial Risk by Jebron Lames (250 points)
edited May 13 by Jebron Lames

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?

1 Answer

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answered May 14 by Pandy (2,770 points)
selected May 24 by Lwazi
 
Best answer
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 :)
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