In this workout, we've been told that an insurance company has written a portfolio of insurance business at the start of year one. And we've been asked to calculate the profits generated in years one to three. Premiums are paid at the start of the year, and claims are settled at the end of each year. Now, the premiums were 100, the loss ratio was 70%, the coverage period was one year and the annual investment return is 10%. We've also been told information as to how the claims are settled over time and that is given relative to the premiums written. So the amounts actually settled in the policy year was 60% of the premiums written. Also, the information is given on a cumulative basis so that by the end of year three, that 70% of premiums written which were paid out as claims is the same as the loss ratio. So we know that all claims were settled by the end of year three. Now, let's start off by calculating our underwriting profits from this portfolio. We start with the premiums, which were 100 in year one and we know that the premiums were nil in years two and three because the coverage period was one year. Now, to calculate the claims expense we're gonna take the loss ratio of 70% and we're gonna multiply that by the premiums for each year. Now, clearly that's gonna give us a claims expense of nil in years two and three but that's okay because the expense is recognized based on when the claims are incurred, not when they're paid. So the underwriting profit is 30 in year one and nil in years two and three. Now, to calculate the investment return, we're gonna need to do an extra calculation that tells us what funds are available for the insurance company to invest. Now, we'll start that with the beginning funds as being equal to the priors ending funds, and the premiums received are BOY, which means they're received at the beginning of each year whilst the claims paid are EOY, which means they're paid at the end of each year. Now, the premiums we can take from above and the claims paid we can take from our claims paid development schedule. So that's 60% of the premiums, so that's claims paid of 60 in year one. And then we sum all that to give the ending funds for each year. Now we can roll forward our beginning funds and premiums received whilst the claim paid in year two, we're gonna do an extra little calculation because we know the amounts only on a cumulative basis and we want to know what was actually paid each year. So we take the 65% and take away the amount from the previous year, and we multiply that whole amount by the premiums of 100. So the claims actually paid in year two were five and we'll do the same for year three.

So that's a 70% less for 65% multiplied by the premiums of 100. So the claims paid in year three were five. So we roll forward our total and we can now calculate our investment return. And we'll take our 10% rate of return and we're gonna multiply that by the beginning funds for the year plus our premiums received. Because we know that those premiums available to invest all year because the claims are paid at the end of each year and then we can roll forward that calculation. Now, clearly the investment return is reducing each year as the amount of funds available to invest decrease because the claims are paid out over time. Now our total profit is our underwriting profit plus our investment return and we now have that for each of the three years.