In this workout, we're told that an insurance company has $20 million of equity and cash. The company writes $180 million of annual insurance policies and immediately invests all the premiums.

We're then asked to build the balance sheet immediately after the premiums are written, and the income statement for the coverage period, using the assumptions provided below.

Now, we can see the 20 million of cash and equity already in the balance sheet, but we need to include the effect of the insurance policies which have been written.

Now, the insurance company will receive $180 million in insurance premiums, and immediately after the business is written, these are all liabilities in the balance sheet, as they reflect unearned premium.

So let's add these in now.

This means that total liabilities and equity, if I add those together, are 200 million. We're also told that the insurance company immediately invests these premiums, so we're going to reflect the premiums received in the investments balance as well. And when we add this to our cash balance, we have total assets of $200 million. So we now have a new balance sheet that reflects the business written and our balance sheet balances, as total assets equal total liabilities and equity of $200 million.

Now let's build the income statement for the coverage year.

I'm going to scroll down for this.

The assumptions provided tell us that the coverage period is one year, the loss ratio is 90%, the expense ratio is 10%, the annual investment return is 6%, and the tax rate is 25%, and all claims are paid out on the final day of the coverage period.

Let's start with the premiums earned.

If the coverage period is one year, then the 180 million of premiums are earned during the year.

If the loss ratio is 90%, then the claims expense is going to be 90% of our premiums earned.

And I'll make that a negative to show it as an expense.

If we deduct the claims expense from the premiums earned, that gives underwriting profit of $18 million.

If the annual investment return is 6%, we can apply this to the insurance premiums received to give our investment return.

Note that we're implicitly assuming that the prior cash balance is being used to cover operating expenses and taxes rather than generating an investment return.

If the expense ratio is 10%, we can apply this to our premiums earned to give our operating expenses.

And I'll multiply that by minus one to show it as a negative.

When we add the investment returns to our underwriting profit and subtract our operating expenses, this gives profits before tax of $10.8 million.

Applying the tax rate to this gives our tax expense of $2.7 million, and we deduct the tax expense from our profit for tax to give net income of $8.1 million.

Finally, the return on equity is net income divided by equity from the balance sheet, giving a return on equity of 40.5%.

Now, this is effectively a simple insurance model, and like all models, we should now spend some time understanding the main drivers.

The first key driver is the growth in business written.

That's the level of premiums. If I increase these to $250 million, let's see what happens to the return on equity.

Gosh, that goes up significantly, as we're now generating more profit, but using the same amount of equity.

Let's undo that change.

The second key driver is product pricing.

That's the premium charged relative to expected claims.

And we can demonstrate this driver in our model by keeping our premium the same, but by decreasing the loss ratio to 85%.

And when I do that, wow, our return on equity goes up, as we're generating more profit, but still using the same amount of equity.

Let's undo that change.

Our third key driver is cost efficiency.

That's the level of expenses relative to the amount of business written.

We can demonstrate this by keeping our premium the same, but decreasing our expense ratio to 8%.

Wow. Again, our return on equity goes up because we're generating more profit, but using the same amount of equity.