In this workout, we're told that an insurance company writes a portfolio of annuity policies at the end of year zero, and provides the following expected premiums and benefits information.

We're asked to use the net premium approach to calculate the insurance reserves, the deferred premium liability, and the premium income and benefits expense each year.

We'll assume that cash flows are as expected and occur at the end of each year, and we're going to use the discount rate provided of 5%.

Let's start with calculating the net premium ratio.

That's the ratio of the present value of expected benefits and claims to the present value of premiums. So we can think of this as the expected loss ratio.

Also, note because these are fixed annuity contracts, all of the premium is received at contract inception. We're going to use the NPV function to calculate the present value of the benefits.

And we'll divide that by the whole of the premium received at year zero.

And that gives us a net premium ratio of 74.4%.

We can then apply the net premium ratio to the gross premium.

So we grab the net premium ratio, lock your cell reference there, and then multiply it by the premiums.

Now, this tells us that of the 500 of upfront premium, 372.2 is needed to cover future benefit payments, with the remaining amount being profit that needs to be deferred to avoid recognizing it all at contract inception.

Now, before we can calculate the deferred premium, we need to calculate reserves, which is the present value of the expected benefits, less the present value of the net premiums. So, we build the calculation again using the NPV function. Now, before we can calculate the deferred premium, we need to calculate reserves, and that's the present value of the expected benefits, less the present value of the net premiums.

And we can build this calculation again using the NPV function.

But notice, that the present value of future net premiums are going to be zero from the end of year zero onwards. So, we actually only need the NPV of the claims.

And that gives me a reserves balance at the end of year zero of 372.2, which is what we should expect, because this is the amount that's received that's needed to cover future claims, and at this point, all the premium has been received.

The next step is to fill in the rest of the reserves movements.

The beginning reserves balance is the prior year ending reserves balance.

We subtract any benefits actually paid, and the benefits expense is a plug.

So unsurprisingly, in year one, the benefits and claims expense is 372.2, and that's the increase in the reserve that year, as no benefits were actually paid.

Now let's roll forward our calculations.

And we can see that the reserves fall each year as benefits are paid out, until they reach zero at the end.

We now need to deal with the deferred premium, as at the moment, we have a claimed expense in year zero of 372.2, and premium actually received of 500.

We need to avoid recognizing all of the policy profit at inception.

So what we're going to do is to defer all of the policy profit and gradually amortize it as the benefits are paid.

Now, this deferred premium is the upfront premium, less the net premium.

So that's 127.8.

We're then going to take this deferred premium, and lock the cell reference, and we're going to multiply this by the reserves balance each year, divided by the initial reserves balance.

Now this obviously starts off at 127.8. That's the full amount of the reserves.

But if we roll this calculation forward, we can see that the deferred premium liability gradually unwinds as the reserves also unwind.

We can then calculate the movement in the deferred premium liability.

So we take the prior year figure, less the current year figure, and roll that calculation forward.

And we're going to include this as an adjustment in the income statement.

So initially, there'll be a large negative adjustment, and then each year, a small positive adjustment as that deferred premium is released.

So we can calculate the premium income as the premium actually received, plus the deferred premium liability movement, and that gives us premium income in the first year of 372.2, which is also the same as the expense on the benefits and claims.

So we get zero profit recognition at inception, as the premium recognized actually matches the benefits and claims expense. But what happens when we roll this forward? In subsequent years, we have premium income which exceeds the benefits and claims expense. And that's so that the profit on the portfolio is gradually released to earnings as the annuity benefits are paid out.