Numerical Example: Surrender value respread to reduce premiums

Consider a very simple endowment policy with sum assured of 100 payable after 4 years. We'll ignore profit, interest, mortality and renewal expenses so that we can do the numbers without tables.

Initial expenses are 20. So the total cost is 100+20=120. This is paid by four annual premiums of 30.

Now, just before paying the third premium, the policyholder wants to increase the sum assured to 200. We decide to calculate the new premium using the surrender value respread method.

First we calculate the surrender value. There's lots of ways we could do this. Let's say we use the asset share, ie 2x30 - 20 = 40.

Then we calculate the premium for a brand new contract. The policyholder wants 200 sum assured and to pay 2 more premiums. To save time, the insurer will calculate the premium using the quotes system, so it still includes initial expenses of 20. So the premium is (200 + 20)/2 = 110

However the premium we charge is the premium for a new contract less the surrender value respread. We respread the surrender value of 40 by dividing by an annuity in advance for two years (which is just 2 in this case as we're ignoring interest and mortality), so the premium actually charged is

110 - 40/2 = 90

However, we'd probably want to adjust this for expenses. We've charged the policyholder for initial expenses of 20 when we used the quaotation system. However the actual expenses will be different to this, eg we won't be paying the same commission as a brand new contract. Say that the actual cost of doing the alteration is only 5. We therefore owe the policyholder 20-5=15 which we could add on to the surrender value to get the premium charged as follows:

110 - (40+15)/2 = 82.5

An important thing to check with each method is whether they are consistent with the boundary conditions. In this case we are looking at the surrender value boundary condition.

If the policyholder reduces the term (keeping the premium the same) then the sum assured will reduce. We would expect that as the term is reduced to zero, then the altered sum assured would reduce to the surrender value (it would be silly if a policy reduced to term of 1 month had a sum assured of less than the surrender value, or considerably more than the surrender value).

In the case of the surrender value respread method, the boundary condition is satisfied because the value we give to the old policy is equal to the surrender value. So if the policyholder wanted to keep the premium the same, but reduce the term to 1 month, then the new sum assured would have to be just a little more than the surrender value.

If a policyholder wants to make their policy paid-up, then we need to solve for the unknown sum assured.

However, the surrender value respread method is used to solve for the unknown premium, so cannot be sensibly used to calculate paid-up values.

This is another boundary condition. If we change a policy to itself, then we should get the same premium.

Consider the example earlier. At time 2, the asset share is 40. A new 2 year policy for the original sum assured of 100 is (100+20)/2 = 60.

So the premium charged (if we refund the full initial expenses to the asset share) is:

60 - (40+20)/2 = 30

This is the same as the original premium, so we have succesfully altered the contract to itself and the boundary condition holds.

However, we could potentially get silly results if the surrender value was calculated on a different basis, eg if we calculated it prospectively on a strong basis and got a surrender value of 60, we would then charge more than 30 for a policy that was unchanged!

Similarly, we'll get a different answer if the premium basis has changed. for example, if interest rates rise (rom the zero assumed here), the premium for the new policy would be lower than 60, and so the premium we'd charge would be less than the original 30.

The features of these alteration methods are hard to follow at first, but I hope these examples help.

Good luck with the exam.

Mark