Directed selection

If insured people are able to choose between different types of annuities at a given provider, those with the shortest life expectancy will choose a front-end guaranteed period annuity, and those with a longer, but shorter-than-average life expectancy will opt for a back-end guaranteed period annuity, while those with a longer than average life expectancy will choose a simple annuity. If we calculate all three types of annuity using an identical mortality table, the result of this selection will be that the insurer makes a profit on the guaranteed period annuities and a loss on the simple annuities, and with a bit of luck the profits and losses of the various annuity types will offset each other.

One way of solving the problem might also be that the annuities are calcu-lated with different mortality tables so that they are adjusted to the mortality of those who, based on rational considerations, voluntarily opt for it. If this is possible, insurers could choose to take advantage of selection instead of strug-gling against it, and in this way they would get a risk community selected on the basis of risk aspects where the clients themselves, using rational considera-tions, voluntarily select the risk community they wish to join. Insurers might even help clients and inform them about which annuity is most beneficial, and if someone is uncertain about how long they might live, insurers could suggest they have a medical check-up, which will help them assess their life expectan-cy. After this, the insurer would simply allow them to make a choice. In this way, risk assessment can be eliminated with the help of precise calculations and “trained advisors”, and what is most advantageous for whom is clearly visible. This also helps to avoid problems resulting from concealing data from assessment.

The principal question is whether this scenario is at all possible, and would such an effort to find a way of operating perhaps lead to contradictions? Be-cause if it turns out that this is not the way to go, precisely the opposite must be done. It is possible for the insurer to lessen the risk to itself by applying the above-described adverse (auto-) selection of annuitants so that the annuities actually provided are drastically diminished compared to the possible annui-ties. If, for example, a provider only sells guaranteed period annuities, it means that its risk will be radically reduced since:

1. A guaranteed period annuity can á priori be provided with a smaller risk than one without a guaranteed period, and what’s more

2. this attracts prospective annuitants with shorter life expectancies (pri-marily men) and makes it less attractive to people with longer life ex-pectancies (mainly women), and in this way the insurer is indirectly se-lecting its own risk community.

Because of the consequences of item 2, whether this possibility should be permitted needs to be seriously considered. It is better if, in this case, the law excludes this possibility and ensures that an annuity provider is obliged to provide all of the annuities permissible by law.

I show below that this solution is not possible, so the targeted selection will not work, and so I conduct an examination, as I did above, into the option of simple, front-end and back-end guaranteed period annuities.

The issue of whether adverse selection owing to choice may be eliminated by using adequate mortality tables might be restricted to a question of whether adverse selection may be eliminated by increasing or reducing the premium, as a client takes into account the different mortality tables only via the premium.

The answer here is not clear, for if we change the mortality table, the pre-mium for the annuity changes along with the composition of those who will opt for such an annuity over others. The issue then is whether this change leads to every type of annuity remaining on the market for an adequate range of clients or whether, in reality, certain types of annuity will lose ground.

In the case of 3 single-life annuities, this question can further be simplified upon examination; can adverse selection loss suffered due to the choice of a simple annuity be eliminated by increasing the simple annuity’s premium or by lowering the premium for guaranteed period annuities? From this point of view, it is of secondary importance whether adverse selection presents itself if the insured can only choose between two types of guaranteed period annuities, and if the answer is “yes”, whether adverse selection might be eliminated by increasing or reducing the premium. As this question is not a practical one, I shall not provide an answer here.

So, the question is: can adverse selection be eliminated by increasing the premium of the simple annuity, or not?

First, I will look at the case of a simple annuity and an annuity with a back-end guaranteed period; and in the case of a net premium, those people will choose a simple guaranteed period for whom it is true that

ä_M[O5|≥ ä

i.e. people who will certainly receive as much in benefit as the amount of pre-mium they pay, since in this case, and only in this case will

ä_M[O5|

ä ≥ä _|+ ∙ ä_M[O5|

ä |+ ∙ ä

be true. If we increase the price of the simple annuity from ä to ä', and in the meantime leave the price of the back-end guaranteed period annuity un-changed (i.e. in the case of the simple annuity we assume a greater remaining

lifetime), then it is obvious that those who have so far chosen a back-end guar-anteed period annuity (i.e. in whose case

ä_M[O5|< ä

was true) would continue buying this because they have no reason whatsoever to change.

In the case of those for whom

ä≤ ä_MNO5|< äe

is met,

ä_MNO5|

äe <ä$|+ v^$∙ ä_MNO5|

ä$|+ v^$∙ ä

will always be true, since ä_MNO5|

äe ≤ 1 <ä$|+ v^$∙ ä_MNO5|

ä$|+ v^$∙ ä

So some clients who chose a simple annuity with a lower premium would now shift to a back-end guaranteed period annuity. In addition, those who change annuities will all be clients in whose case the received benefit will be smaller than the premium paid for the simple annuity; simply annuities will continue to be only purchased by people who receive a bigger payout than the premium of the simple annuity.

Based on the above, we can ascertain that if we increase the price of a sim-ple annuity, we exclude clients from buying a simsim-ple annuity who would have chosen it at a lower price, but for whom the new, higher price exceeds the anticipated benefit. Only those clients for whom the anticipated benefit will again be higher than the price will choose the simple annuity with the new price, while others will convert to buying a back-end guaranteed period annui-ty. This means that the simple annuity continues to make a loss, so raising the price of the simple annuity will not result in simple annuities becoming profit-able for the provider, and such attempts would eventually lead to simple annui-ties being edged out of the market.

Assuming a gross premium does not significantly change the above; the logic applied with regard to the net premium can also be applied here, so ad-verse selection cannot be eliminated by increasing the premium.

Examining simple and front-end guaranteed period annuities in the net premium case (although assuming gross premiums will again not change the logic in this case), based on the above the threshold between the simple and the front-end guaranteed period annuities is the

ä_M[O5|< ä _|∙ ä

ä |+ ä _|

inequality (where h < g will obviously also be true and the relationship ä_M[O5|< ä

will also be valid). If the inequality is true, the client will choose a front-end guaranteed period; if it isn’t they will choose a simple annuity.

It is obvious that the best simple annuity clients would choose the front-end guaranteed period annuity, so the simple annuity will generate a loss, although not to the extent we might see in the case of simple and back-end guaranteed period annuities, because in this case the simple annuity will also be purchased by people for whom the received benefit will be smaller than the premium paid. The question is whether the profitability of the simple annuity may be restored by increasing the premium of the simple annuity, i.e. if we increase the premium from ä to ä'.

In the context of an increasing premium, we must immediately ask whether there is some cap on it. It is obvious that the premium of the simple annuity cannot be increased above the premium of the front-end annuity as there would be no argument in favour of buying a simple annuity instead of a front-end guar-anteed period annuity in such a case. In fact, we can be a little more precise: the premium of the simple annuity must be lower than the premium of the guaran-teed period annuity as this is the sole reason the client chooses an annuity that offers a lower benefit from among two annuities with the same premium.

If the premium of the simple annuity were raised to the level of one with a front-end guaranteed period, and if we were to suppose (in contrast to the above) that only those individuals purchase a front-end guaranteed period annuity for whom the guaranteed period actually guarantees something on the basis of their remaining lifetime, then we may state that in the case of a pro-vider that offers two (i.e. simple, and front-end guaranteed period) annuities, all adverse selection could be eliminated. Under such conditions, the provider actually only sells an annuity with a front-end guaranteed period while in the case of an annuity the problem of adverse selection owing to choice cannot emerge, or at least not at the level of the provider. So the insurer collects pre-mium from the entire risk community and it pays out the same sum in benefits.

However, the clients – also based on the above – are divided among the two formally different types of annuity whereby individuals whose life expectancy remains below the guaranteed period choose the guaranteed period annuity, while the rest opt for the simple annuity. In these circumstances, people who choose a guaranteed period annuity obviously pay more in premium than they receive as benefit, and therefore those who opt for a simple annuity obviously

pay less if the entire risk community is in balance with respect to premium and benefit. So the loss generated by the simple annuity cannot be eliminated by raising the premium.

The question arises whether the reverse could provide a solution, i.e. can we stop adverse selection by lowering the premium of guaranteed period annuities? In other words, the question is whether the range of simple annuity buyers will grow in response to an inverse strategy i.e. in response to decreas-ing the premium of either of the (front-end or back-end) guaranteed period annuities. The answer is obviously no, because for those individuals for whom it was worth buying a guaranteed period annuity to begin with it will have become even more worthwhile. In addition, a segment of simple annuity buy-ers would shift to becoming guaranteed period annuity buybuy-ers, i.e. those with the shortest remaining life expectancy from among the simple annuity buyers.

But relatively speaking, these were the most favourable clients from a simple annuity perspective, so the simple annuity might generate an even higher loss, meaning the loss cannot be eliminated in this manner.

In summary: in a given situation where clients know their remaining lifespan and all of them wish to maximise the total benefit received compared to the premium paid, it would be impossible to see the single annuity as profit-able in itself as neither increasing the premium of a single annuity nor decreas-ing the premium of some of the guaranteed period annuities would prevent the single annuity maximum from losing ground on the market. In other words, this means that it is impossible to prepare selection tables that reflect the mortality of individuals who opt for different annuities, and therefore directed selection does not work as a solution.