General considerations – loading thresholds in gross premium cases In advance, and very generally, the following can be said about loading:

• In the case of different annuities, the relative value of the ^J_K ratio does not change if the loadings are the same, that is if λ^X= λ^Y= λ^N,⁵⁴ then the rule of selection among different annuities will be identical in the case of both gross and net premiums.

• Since the client receives the most benefits from a back-end guaranteed period, then from a front-end guaranteed period and finally from a sim-ple annuity (provided that g is identical for both guaranteed period an-nuities and is not 0 or 1), the following inequality must be true with re-spect to the gross premium:

!ä _|+ ∙ ä" ∙ 1 + Z^[ > !ä _|+ ä _| " ∙ 1 + Z^\ > ä∙ 1 + Z^] Of this latter inequality, certain thresholds are added on top of the possible magnitude of loadings of different annuities; though the problem of thresholds may be raised more generally, and issues relating to different types of

54Loadings for different annuity types: λ^X for a simple annuity, λ^Y for an annuity with a front-end guaranteed period and λ^N for an annuity with a back-end guaranteed period.

old arise in the case of gross premium (contrasting with the problem of the net premium). This problem does not manifest itself in connection with the net premium and the only question to be answered is which of the two or three annuities are worth getting if one knows one’s remaining lifetime and it is taken for granted that a person is interested in maximising all the benefits received from the provider. If we assume that everyone knows how long they will live, the question of choice only makes sense if we presume that clients are being obliged to purchase an annuity from the money they have accumu-lated for that purpose. Should this not be the case, a reasonable solution for many would be to not buy an annuity at all but to budget for their remaining lifetime. However, if we disregard the assumption that there exist “all know-ing” annuitants, then in the case of net premium the idea that one of the annui-ties is not worth getting (in general, or in comparison with other options) does not arise, not only because buying an annuity is mandatory, but also because the annuity calculation is “fair”, i.e. it is based on an equation between the premium and the expected value of the benefit.

This is not the case with the gross premium, where three further questions may be raised with respect to the absolute and relative magnitudes of the cost (loading) portion of annuities, this being in addition to the above-discussed problems related to the net premium:

I. Is it worthwhile for anyone to buy one of the annuities i.e. does it have an excessively large loading (apart from the fact of whether or not it is mandatory to buy some form of annuity)?

II. Is it worth purchasing the given annuity type in comparison to the other annuity types, i.e. does it have excessive loading compared to others?

III. From what loading value is it worth purchasing one annuity compared to another for a client with a given length of lifetime remaining?

In the first two cases, the investigation concerns the “final” limits, i.e. is there anybody at all existing among eventual clients with a different remaining life-time (who were the same age at point of entry) for whom it is worth buying one given type of annuity? In the third case, the question is more specific, therefore the threshold of type III must be within the thresholds of type II, and type II within that of type I. Below, I investigate the size of threshold types I and II, and will later use this for delineating the thresholds for type III.

These three issues have been raised because of the need to include loading in the investigation – and its magnitude obviously has rational limits; the ques-tion is where these limits are.

If we first examine type I thresholds and the definition of the threshold up to which premium limit it is worth purchasing an annuity, we can see that the

criteria for defining such a threshold in this case is not completely obvious.

Nevertheless, it may be logical to say that if the maximum possible benefit from an annuity is smaller than its premium, it is not worth obtaining. Alt-hough if it is mandatory to buy an annuity, the issue is not whether or not it is worth buying a specific annuity, but it is still a good idea to be aware of the theoretical upper price limit.

The possible maximum benefit of the annuity is not self-evident as its value depends on how long the annuitant with the longest life is going to live. Natu-rally, nobody can say, so we can only define a theoretical value. Thus, in order to calculate the maximum possible benefit, the (statistically relevant) maxi-mum possible age must first be decided and also by what discount coefficient we shall reduce the annuity units to a common denominator. In my view, the logical answer to both questions is “the coefficient that was taken into account in the calculation”. So the maximum age for the mortality table should also be the highest age here, though we do know that there is a (small) chance of someone having a higher actual age than this as we do not really know what the “biologically possible” maximum age is. For discounting, it is reasonable to take into account the technical interest rate used in the calculation as a client would also be able to achieve that if they were managing their own assets and didn’t purchase an annuity.

From this, the following inequalities may be established, and from these we may determine the threshold of type I:

• The possible maximum benefit of a back-end guaranteed period annuity:

1 + + ^;+ ⋯ + ^>=+ ^>=5+ ⋯ + ^>=5 =1 − ^>=55

= ä_{>=55 |} 1 −

so for the gross premium it must be true that:

ä>=55 |> !ä |+ ∙ ä" ∙ 1 + Z^[

and via this the rule applicable for the loading maximum is:

ä_{>=55 |}

ä |+ ∙ ä > 1 + Z^[

• The possible maximum benefit for a front-end guaranteed period annuity (assuming ω-x>g, without which we would not, in effect, be referring to a life annuity but an annuity certain) is ä_{^-5_}; it must hold true for the gross premium that:

ä_>=5|> !ä _|+ ä _| " ∙ 1 + Z^\

From this, the rule applicable for maximising the loading is:

ä>=5|

ä _|+ ä _| > 1 + Z^\

• The possible maximum benefit from the simple annuity is the same as that of the front-end guaranteed period, so for the gross premium it must be true that:

ä_>=5|> ä∙ 1 + Z^]

And via this we have a rule applicable for the maximum loading:

ä_>=5|

ä > 1 + Z^]

In answering question II, the observation made at the beginning of this section gives us some support. According to this, there is a strict hierarchy of benefit among three different single annuities (if we assume that g is the same for two guaranteed period annuities and is at least 2); the most is provided by a back-end guaranteed period annuity, the next highest by a front-back-end guaranteed period annuity, and the least of all is provided by a simple annuity. So the following inequality must exist with respect to gross premiums:

!ä_$|+ v^$∙ ä" ∙ !1 + λ^N" > !ä_$|+ ä_$| " ∙ 1 + λ^Y > ä∙ 1 + λ^X In other words: if these inequalities do not exist it is not worth the client buy-ing certain types of annuity so the loadbuy-ing of a simple annuity compared to the loading of the front-end guaranteed period annuity cannot exceed a certain level, otherwise only the front-end guaranteed period annuity is worth purchas-ing; and this situation is the same with respect to simple and back-end guaran-teed period annuities.

Further progress as regards the question can be made by exploring the bene-fit hierarchy amongst annuities:

• From two annuities, the one with a higher benefit will be bought by the client if its gross premium is not obviously higher than the gross pre-mium of an annuity with a lower level benefit, i.e. if the loading is too small compared to the loading of the other annuity.

• However, nobody will purchase the higher-benefit annuity if the gross premium is much higher than the provided excess benefit, i.e. if the loading is too high.

The above inequalities assist in determining a loading that is “too small”. Let us look at these relative lower limits; if any of the previous three inequalities are not satisfied, we may be certain of the result.

• if !ä_$|+ ä_$| " ∙ 1 + λ^Y ≤ ä∙ 1 + λ^X, and in fact even if !ä_$|+ v^$∙ ä" ∙ !1 + λ^N" ≤ ä∙ 1 + λ^X, then nobody will purchase a simple an-nuity

• if !ä$|+ v^$∙ ä" ∙ !1 + λ^N" ≤ !ä$|+ ä$| " ∙ 1 + λ^Y, then everyone will purchase a back-end guaranteed period annuity instead of a front-end guaranteed period annuity.

Therefore I assume that:

!ä_$|+ v^$∙ ä" ∙ !1 + λ^N" > !ä_$|+ ä_$| " ∙ 1 + λ^Y > ä∙ 1 + λ^X will be true, which may also be stated in these alternate forms:

1 + λ^Y

1 + λ^X > ä

ä$|+ ä_$|

and

ä$|+ v^$∙ ä

ä$|+ ä_$| >1 + λ^Y 1 + λ^N or differently,

1 + λ^N

1 + λ^Y > ä$|+ ä$|

ä_$|+ v^$∙ ä

It is naturally true that

1 + λ^N

1 + λ^X > ä

ä$|+ v^$∙ ä

but this is already a consequence of the previous two.

Further considerations are required for an exploration of “too large” loadings.

It is obvious that the premium for “dominant” annuities (i.e. the other two regarding simple annuity, and back-end guaranteed period annuity in the case of the front-end guaranteed period annuity) can be increased to such an extent that buying them (in comparison with other annuities as alternatives) becomes á priori irrational for a client. Calculation of the threshold can begin from the ^J_K quotient, so let us examine the question according to annuity pairs. In the ex-amination I have utilised the observation that it will be in no one’s interests to

purchase an annuity of higher benefit (what we referred to above as “domi-nant”), if it is not worth purchasing by those who can benefit the most from this higher level of service compared to annuities that provide less benefit. So:

• With respect to simple and back-end guaranteed period annuities, the question of when it will never be worthwhile getting a back-end guar-anteed period annuity may be seen in the light of when it will always be true that:

ä∙ 1 + λB ^X > B

!ä$|+ v^$∙ ä" ∙ 1 + λ^N

Clearly it is true that the most difficulty occurs if the benefit is a mini-mum, as in this case the back-end guaranteed period annuity would normally be better for the client. An extreme case of this is when a cli-ent dies immediately, so they only receive EUR 1 annuity in the case of a simple annuity and ä_$5| annuity in the case of the back-end guar-antee. This is when the difference in the benefits of the two annuities will be maximum. So the issue is when the following inequality be-comes true?

1

ä∙ 1 + λ^X > ä_$5|

!ä_$|+ v^$∙ ä" ∙ 1 + λ^N Clearly if:

1 + λ^N

1 + λ^X > ä∙ ä$5|

ä_$|+ v^$∙ ä

Thus it can be stated that the inequality ä∙ ä_$5|

ä_$|+ v^$∙ ä≥1 + λ^N 1 + λ^X must be true.

• Regarding simple and front-end guaranteed period annuities, the annui-tant will not take out a front-end guaranteed period annuity for the same reasons (as the difference in benefits between the two annuities is big-gest in connection with annuitants who have the shortest life expectan-cy), if:

1

ä∙ 1 + λ^X > ä_$|

!ä$|+ ä_$| " ∙ 1 + λ^Y

In other words:

1 + λ^Y

1 + λ^X > ä∙ ä_$|

ä_$|+ ä_$|

So the inequality

ä∙ ä$|

ä$|+ ä_$| ≥1 + λ^Y 1 + λ^X must always come true.

• The difference between the benefits of the front and back-end guaran-teed period annuities gradually increases in the first g year, then it re-mains constant after the expiry of the front-end guaranteed period; i.e.

at present value it is biggest in real value in year g. So the question is when does the inequality below:

ä$|

!ä$|+ ä_$| " ∙ 1 + λ^Y> ä$5$|

!ä_$|+ v^$∙ ä" ∙ 1 + λ^N

= ä_$|+ v^$∙ ä_$|

!ä$|+ v^$∙ ä" ∙ 1 + λ^N become true?

Or rather:

1 + λ^N

1 + λ^Y>!ä$|+ ä$| " ∙ 1 + v^$ ä$|+ v^$∙ ä

So the inequality

!ä_$|+ ä_$| " ∙ 1 + v^$

ä_$|+ v^$∙ ä ≥1 + λ^N 1 + λ^Y must always come true.

In summary, the threshold values are:

The “absolute” maximum of loading, i.e. the threshold values of type I, are:

• in the case of back-end guaranteed period annuities:

ä^=55$|

ä$|+ v^$∙ ä> 1 + λ^N

• for a front-end guaranteed period annuity:

ä_^=5|

ä$|+ ä_$| > 1 + λ^Y

• for a simple annuity:

ä_^=5|

ä > 1 + λ^X

It is of course logical to assume that the costs are positive, so 1 + λ^N> 1, 1 + λ^Y> 1 and 1 + λ^X> 1 become true, but this has no significance going forward.

The relative values of the loading must fall between the following extremes (type II threshold values):

ä∙ ä_$5|

ä_$|+ v^$∙ ä≥1 + λ^N

1 + λ^X> ä

ä_$|+ v^$∙ ä

ä∙ ä_$|

ä_$|+ ä_$| ≥1 + λ^Y

1 + λ^X> ä

ä_$|+ ä_$|

!ä_$|+ ä_$| " ∙ 1 + v^$

ä_$|+ v^$∙ ä ≥1 + λ^N

1 + λ^Y> ä_$|+ ä_$|

ä_$|+ v^$∙ ä

As it certainly becomes true that

ä_$|+ v^$∙ ä> ä_$|+ ä_$| > ä

(at least if the guaranteed period is effective, i.e. g≥2), thus it will be true for every type II lower threshold value that it is smaller than 1, i.e.:

1 >1 + λ^N 1 + λ^X 1 >1 + λ^Y 1 + λ^X 1 >1 + λ^N 1 + λ^Y

The question is what will the criteria be for choosing between annuities within the above thresholds? And can we give a more specific answer to the loading values while also taking into account the remaining lifetime of the client?

3.4.2.2. The choice between simple and back-end guaranteed period annuities First, I shall investigate the case of the net premium. The question is; when does the inequality

ä_MNO5|

ä ≥ ä_MNO55$|

ä_$|+ v^$∙ ä

become true – and when is its inverse true?

The result is that if

ä_MNO5|≥ ä

is true then the client will choose a simple annuity rather than a back-end an-nuity.

Since we can (approximately) call ä_MNO5| the “discounted remaining life-time”⁵⁵ and ä the “discounted average remaining lifetime”, we can say in summary (and somewhat liberally) that the threshold for choosing between a back-end guaranteed period and a “simple” annuity will always be the dis-counted general remaining lifetime, irrespective of the magnitude of the guar-anteed period. Independently of the duration of the guarguar-anteed period, the client will always choose a “simple” annuity in the case of a discounted re-maining lifetime above this, and below it the client always chooses a back-end guaranteed period annuity, provided the calculation is of the net premium and both variations were prepared using the same mortality table. Here, the “sim-ple” annuity will certainly generate a loss, and the back-end guaranteed period annuity will certainly be profitable for the insurer.

This choice is presented in the diagram below:

55Since with a 0% technical interest rate, ä_MNO5| can be simplified to MhO + 1 and ä

to e+ 0.5, the dividing line will be the inequality MhO + 1 ≥ e+ 0.5. This may be otherwise illustrated as MhO ≥ e-0.5. This is a similar, although not identical rule to the one which states that the remaining lifetime must be higher than the expected remaining lifetime.

Diagram 2: The annuitant’s choice between “simple” and back-end guaranteed period annuities

"simple" annuity

back-end guaranteed period annuity

g = maximum possible guaranteed period

h = the remaining life span

This is where the remaining average, discounted life span of the insured - äx - first reaches the discounted remaining life span - ä[h]+1 .

If we extend our analysis to the case of the gross premium, we need to ask when the

ä_MNO5|

ä∙ 1 + λ^X ≥

ä_MNO55$|

!ä$|+ v^$∙ ä" ∙ 1 + λ^N inequality is true, and when its inverse is true.

Obviously, this is if

ä_MNO5|≥1 + λ^X 1 + λ^N∙ ä

is true.

So with the gross premium the net premium rule is modified, and the above line is shifted by the ratio of the loading, i.e. above the

ä_MNO5|=1 + λ^X 1 + λ^N∙ ä

line, the client chooses a simple annuity, while below it they will choose a back-end guaranteed period annuity.

The limits discussed above must also be considered with the loadings.

In document Model Options for Mandatory Old-Age Annuities (Pldal 130-140)