In this workout, we've been told that an insurance company has written some portfolios of insurance business in years one to three, and has ceded some of the risk to a reinsurance company, as detailed below.

Now, we've been asked to calculate the underwriting profits generated in years one to three. In terms of the information we've been given, we've been told that the premiums written, which is the premiums received from the customers, was 100 in each year. But we've also been told that the reinsurance ceded was 15 in each year, and that means that the premiums paid to the reinsurance companies were 15 each year.

Next, we've been told the loss ratio for each of the three years, and also that the coverage period for the policies is one year.

Finally, in terms of the reinsurance cover, we've been told that these are stop-loss reinsurance policies, and that means that portfolio claims in excess of a certain threshold are clawed back from the reinsurer. And in this situation, the threshold is 60% of premiums. Now, let's start off by calculating the premiums, and we start with the gross written premiums, which were 100 in each year.

And we deduct from that the reinsured premiums, so the premiums paid to the reinsurers, and we can use that to calculate the net written premiums, which is the net amount of premium received by the insurance company.

Now, to calculate the claims expense, we're going to start with the gross claims expense, which is the loss ratio for each year, multiplied by the gross written premiums.

Now, to calculate the reinsurer's share, we need to remember that this is a stop-loss reinsurance policy, and therefore, all claims in excess of 60% of premiums will be clawed back from the reinsurer.

Now, we're going to use a max function to calculate the reinsurer's share, because we need to make sure that it's always a positive amount which is clawed back from the reinsurer, even if the claims are below the threshold. So we use a max of zero and the claim amount. But we need to show this as a negative number, so we'll do minus the max of zero and the claims that we can claim back, which is the gross claims, less the threshold of 60%, multiplied by the gross written premiums.

And that gives us a reinsurer's share of 10, and we can roll forward that calculation for each of the three years.

Now, the net claims expense is just the gross claims expense, less our reinsurer's share.

Now, you'll notice that the net claims expense is 60 in each year, even though the gross claims expense changes each year.

Now, does that make sense to us? Well, given that this is a stop-loss reinsurance policy and all claims in excess of 60, that is 60% of gross written premiums, are claimed back from the reinsurers, then absolutely it makes sense that the net claims expense shows us 60 in each year.

Finally, we can now calculate our underwriting profit, which is our net written premiums, less our net claims expense, and that gives us an underwriting profit of 25 in each year.