In this workout, we've been told that an insurance company has written some portfolios of insurance business throughout 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. Now, the premiums written, which is the premiums received from the customers, was 100 in each of the three years.

But we also know that there was some reinsurance ceded, and that reflects the amount of premiums paid away to the reinsurance companies, and that was 15 in each of the years. Next, we've been told the loss ratio in each of the years, which can help us to calculate the claims expense. Now, in terms of the reinsurance cover, we've been told that this is a proportionate reinsurance cover, and that means that the insurance company can claim back a fixed percentage of claims from the reinsurance company. And in this situation, it's 15%.

Finally, we've also been told that the coverage period for all the policies is one year. So let's start by calculating the premiums, and we start with the gross written premiums of 100 in each year. We then deduct from that the amount of reinsured premiums of 15 in each year.

And this allows us to calculate the net written premiums, that is the net amounts received by the insurance company.

We can now calculate the claims expense in a similar way, and we start off with the gross claims expense, which is the loss ratio for each year, multiplied by the gross written premiums.

Now clearly, because the loss ratio isn't the same in each year, we have a different claims expense. So the 70% loss ratio in years one and three gives us a claims expense of 70, whilst the 75% loss ratio in year two gives us a claims expense of 75.

Now from this, we know that the insurance company can claim back 15% of those claims from the reinsurance company.

So let's take that 15% and multiply it by the gross claims expense to give the reinsurer's share.

Now the net claims expense, that is the net expense to the insurance company, is simply the gross claims, less the reinsurer's share.

Now, the underwriting profit is just the net written premiums, less the net claims expense, giving 25.5 in year one, 21.3 in year two, and 25.5 in year three.