Most books, including the abovementioned ones, usually present the mathe-matics of annuities by explaining how certain types of annuity can be con-structed out of smaller units (certain or conditional lump sum payments with differing conditions) and this is the basis for the various relationships between the premiums of concrete annuity types and other life insurances or financial
products. In this book I will follow a different, precisely opposite logic, that I have developed myself. I do this firstly because it makes the relationship be-tween different types of annuity clearer and secondly because as a result I can present a “global”, “birds eye view” of annuities without concentrating on the technical details (which are presented well by the above-mentioned works).
Following this logic, I begin with presenting the relationships between the premiums of various annuities.
1.2.1. THE RELATIONSHIPS BETWEEN THE NET PREMIUMS OF VARIOUS ANNUITIES AND OTHER LIFE ASSURANCES
It may be stated that just as the light of the sun can be split up into all the col-ours of the rainbow, and just as white light contains all colcol-ours, so the simplest annuity, perpetuity, contains all possible annuity types and the other financial products related to annuities. Below, we will see how they form a part of it and how the various constructions are related to each other. I restrict my analysis (for the sake of simplicity and clarity) to annuities due, but with minor modifi-cations this calculation can also be used for annuities payable in arrears.9 The appropriate, standardised notations will be introduced according to the order of explanation (and are listed at the end of the book).
“In the beginning there was the perpetuity”, we might say. The simple, stand-ardised form of perpetuity is when somebody receives i interest at the end of each year on his/her EUR 1 capital and spends it. In this way the magnitude of the principal remains (nominally)10 unchanged for eternity, as does the result-ing income, which in view of its regularity we can regard as an annuity. The payments of this annuity are always due at the end of the year, so this is an
9The distinction between annuities-due and annuities payable in arrears is quite technical, and from the perspective of the formulae it is not particularly important. The point of the difference is that the particular payments are due at the beginning (annuity-due) or at the end (payable in arrears) of the intervals between two payments. It is obvious that the two annuities differ from each other only in the first (and perhaps the last) payments, the rest of the payments are the same. From the point of view of calcu-lations it is important to know the exact situation. We denote the net (without costs) single premium as „ä” in case of annuity-due and as „a” in the case of annuity payable in arrears. Our choice also means that in the following we use only the variation ä, instead of duplicating the (very similar) formulae.
10Naturally, our analysis can be relatively easily extended to principal unchanged in real terms. In this case, we must split the nominal interest rate into two parts and the real interest rate will play the same role as the nominal interest rate in this analysis. In practice, land rent as a kind of annuity is the closest to perpetuity and represents a more or less unchanged principal in real terms.
annuity payable in arrears. How we can turn this EUR 1 principal and i interest rate into an annuity-due perpetuity? The question can be reformulated: what sum can we deduct from the EUR 1 capital at the beginning of the year, so that with an i rate of interest rate the principal will again amount to EUR 1 at the end of the year? If we denote (as is usual) this sum with d, then we receive the following equation:
1 − d ∙ 1 + i = 1 from which:
d = 1 − 1 1 + i
The usual practice is to denote the reciprocal (1+i) with v, the so-called dis-count factor:
v = 1 1 + i So:
d = 1 − v from which firstly:
v = 1 − d and secondly:
d = 1 − 1 1 + i = i
1 + i = i ∙ v
So d is the discounted value of i, and this is indeed logical, because the ques-tion could also have been formulated as follows: what rate of interest rate would we get instead of i, if we want to receive our payment a year earlier?
The answer, naturally, is the discounted value of i, i.e. i ∙ .
So we have our first standardised annuity-due, where the annual payment (in advance) on EUR 1 principal at an i interest rate is d. The notation of the single premium of the perpetuity (for annual EUR 1 payment in advance) is:
ä|
So, the equation of a perpetuity with annual payment d is
1 = d ∙ ä|
This expresses the fact that for EUR 1 the client receives an eternal annual cash-flow of d magnitude always in advance at the beginning of the year.
Below, when I write an equation with 1 on one side of it, it expresses what products the client can receive for EUR 1, which is logically equivalent to a single perpetuity part.
Before we continue, it is instructive to express ä| from the above equation:
ä|=1 = 1
∙ =1 +
= 1 +1
where is the well-known formula for a unit of perpetuity payable annually in arrears (a|). Annuity-due perpetuity to all intents and purposes differs from this only with respect to the EUR 1 paid right at the beginning of the annuity period.
So perpetuity is a cash-flow that lasts for eternity. Naturally, we can “sever”
this cash-flow in different places during its lifetime. What happens, for in-stance, if we only wish to receive EUR d in advance annual annuity-due pay-ments on our EUR 1 capital until the time our death? Then obviously immedi-ately following our death (i.e. at the next payment due date) our heir11 will get back the EUR 1 principal (which he/she may again invest in a perpetuity if they so wish), because the point of perpetuity is that the whole original capital is always “restored” again and again one year after the interest payment. But from another perspective, this means that we have split our uniform perpetuity into two financial products:
1. An annuity-due lasting until our death with annual payment d, i.e. a life annuity, and
2. A life assurance with an assured sum of EUR 1 and payment due at the time of death (a whole life policy).
We can purchase these two financial products for exactly EUR 1. The usual notation of the net single premium of an annuity-due life annuity with an an-nual payment EUR 1 is ä, where x is the entry age of the annuitant (until whose death the annuity lasts) at the commencement of the contract, while the notation of the net single premium of a whole life assurance with EUR 1 sum
11I am consciously not dealing with the fact that the insurer pays the sum assured to the beneficiaries independently from the probate process, because legally, if there is a formal beneficiary then the sum assured does not become part of the inheritance. It nevertheless constitutes an inheritance to all intents and purposes.
assured is . In both cases the implicit assumption is that the financial institu-tion (in this case a life insurer) invests our capital at an annual interest rate of i and does not charge anything for its service.12 Using these notations we can write down this “segment” of the perpetuity, as:13
1 = ∙ ä+
Naturally we can sever the cash-flow of the perpetuity at other places too.
We also distinguish between two other important cases of standardised annui-ty, and especially with reference to life annuities:
1. The annuity lasts for a predetermined term (for the sake of simplicity for a certain number of whole years), after which the capital is paid out, and
2. The annuity lasts for a predetermined term, but maximum until the time of our death, after which the capital is paid out (or transferred to our heir[s]).
In the first case it must obviously be necessarily true that 1 = ∙ ä|+
where I denote the (whole) number of years of the term with n, the net present value of an annuity certain with an annual premium of EUR 1 with ä|, and the present value of 1Ft due in n years with vn.
While in the second case it is clearly true that
1 = ∙ ä:|+ :|
where ä:| is the net present value of the EUR 1 annual life annuity-due until death, but maximum for n years, while A:| is the net single premium of a life insurance for death and maturity (an “endowment”) with EUR 1 sum assured.
Above, we first severed the perpetuity when someone died. This event can be logically extended to more than one (two in the simplest case) deaths. Then we get the following equation (with two clients insured):
12Or at least it makes to do with the “margin” between the interest it actually earned and the interest it pays to the client.
13Most of the actuarial books include this equation. See e.g. Bein-Bogyó-Havas p182, Bowers et al. p131, Gerber p36, Krekó p30, Neill p63, but a significant number of the equations shown here are missing.
1 = ∙ ä+
where ä is the net single premium of a EUR 1 annual annuity-due for two insured persons that lasts until both are alive. (They are x and y years old, respectively, at entry, or below, for the sake of simplicity, annuitants x and y).
A is the single net premium of a whole life insurance with two insured per-sons, which pays EUR 1 when either of annuitants x or y dies.
Naturally, joint life annuities also have a temporary variant, in which case the equation
1 = ∙ ä:|+ :|
will be true, where A:| is the net single premium of a joint life endowment that pays a EUR 1 sum assured when one of the insured persons dies or after a period of n years, while the ä:| single premium joint life annuity lasts until both annuitants are alive and n years have not yet passed.
A further extension is the addition of a guaranteed period (g years) to the annu-ity. The following can be stated with regard to the practical reasoning behind these. In the case of the immediately commencing and non-conditional annui-ties analysed so far it can happen that the annuitant dies not long after conclud-ing the contract and the annuity payments cease immediately after the first is paid out (or in the case of conditional or deferred annuities might not begin at all). This inevitably scares off many potential annuity buyers from purchasing an annuity, who vacillate between spending their accumulated capital and leaving it to their children, and makes this largely dependent on their life ex-pectancy, which they cannot know in advance. By guaranteeing payment of the (usually lifelong) life annuity for a number of years, providers can reduce this psychological barrier to the purchasing of annuities. The guaranteed period can be included at the beginning (front-end) or at the end (back-end) of the term. In case of an annuity with a guaranteed period at the beginning (which we will denote with | ä), the annuitant (or their heir) is guaranteed to receive the payments due for the first g years of the annuity even if they die in the mean-time. Naturally, this guarantee will have no effect if the annuitant dies after g years.
An annuity with a guaranteed period of this kind obviously provides more than an annuity with no guaranteed period, so its premium is also higher. Ac-cordingly, EUR 1 is no longer enough (above the annual d payment) to cover the previous whole life insurance, only a modified form of it. This modified form means that the EUR 1 benefit will certainly not be paid during the first g years, even if the insured party dies during this period. If the insured party dies
during the first g years, the beneficiary will also only receive the EUR 1 at the end of the g year period. This is as if the first g years represented a unit, and the annual consideration of mortalities only commences once this period has expired. Let us denote this with a g in the lower left hand corner (where g means the length of the first such period). Then we can write the following equation:
1 = ∙ ä| +
In the case of an annuity with a guaranteed period at the end (which we will denote with |ä), the beneficiary will receive payments for g years following the death of the insured party. This obviously means that the sum assured of the whole life insurance will also be payable for g years following the death, i.e. we “prolong” payment of the death benefit for g years. This can be express by saying that in case of death beneficiaries are not yet due the EUR 1 unit, but only its value discounted by g years, so our equation will change as follows:
1 = ∙ ä | + ∙
The parts of the above equations can be further segmented. The following equation is obviously true:
ä= ä:|+ ä|
i.e. the lifetime annuity can be broken up into an n-year (n ‹ ω) temporary annuity-certain and a deferred (life) annuity with deferment period n. Accord-ingly, the equation 1 = d ∙ ä+ A can be also written in the following form:
1 = ∙ !ä:|+ ä| " +
(A deviation from the basic formula is that in this case not every insurance will necessarily result in the payment of benefits, because the deferred annuity ceases without payment if the insured individual dies within the n years.)
Analogously to the annuities A can also be split into a “temporary” and a
“deferred” sum assured. The “temporary” insurance is the classical term (death) insurance, and we traditionally denote deferred insurance with#| A ! (Remarks: 1. I must emphasize that I have denoted the period with m and not n, because the length of deferment does not necessarily have to be the same as in the case of annuities, and 2. #| A should not be confused with the previous-ly introduced$ A , which means something different. In the first, the vertical denotes a condition, i.e. a contingent benefit [if the insured person dies within
the first m years then no payment is due according to this insurance policy], while in the second case there is no condition, we have simply “combined” the first g years.)
Accordingly, it is also true that
= :%|+ %|
where A:#| is the traditional notation of the single premium of a term (death) insurance with EUR 1 sum assured and a term of m years. So we can break up our basic equation 1 = d ∙ ä+ Afurther into the following form:
1 = ∙ !ä:|+ ä| " + :%|+ %|
(In this formula either the classical terminsurance or the deferred whole life insurance will definitely cease without paid benefit, but it is possible that the deferred annuity will also not involve the payment of benefit.)
Although I do not deal with the topic, I must remark that
1. Temporary annuities may also be broken up into the sum of a short temporary and a deferred temporary annuity,
2. Joint life annuities with two or more insured persons can also be broken up similarly to single client annuities.
Annuities with a guaranteed period are complex products so they may natural-ly also be broken up.
A front-end guaranteed period annuity is clearly the sum of an annuity-certain and a deferred annuity, that is
| ä
= ä |+ ä |
So the equation 1 = d ∙ ä|$ + A$ changes as follows:
1 = ∙ !ä |+ ä | " +
Back-end guaranteed annuities provide a greater guarantee than front-end guaranteed annuities, because:
1. They include the guarantee provided as front-end guaranteed annuities, since annuity payments are also due during the first g years in this case, and
2. In contrast to front-end guaranteed annuities, payments will definitely last longer in this case than without a guaranteed period. In the case of a front-end guaranteed annuity, the guarantee is not effective if the
in-sured party dies after the guarantee period, but in the case of a back-end guaranteed annuity there is no such break.
Back-end guaranteed annuities, as a complex product, may be interpreted in two ways:
1. It is to all intents and purposes an annuity certain with a term of g years (because the insured party is certain to receive payments during the first g years, even if they die immediately after purchasing it) and a “normal” life annuity whose payments are all “shifted” by g years, meaning:
|ä
= ä |+ ∙ ä
This interpretation is totally compatible with the “shifted” whole life insurance in the equation 1 = d ∙ ä$| + v$∙ A. In this case, the equa-tion receives the following form:
1 = ∙ !ä |+ ∙ ä" + ∙
or
1 = ∙ ä |+ ∙ ∙ ä+
2. It is naturally a normal lifelong annuity plus such a whole life insur-ance, where the sum assured is paid as an immediately commencing annuity certain with a term of g years, so:
|ä
= ä+ ä |∙
In this case, the equation 1 = d ∙ ä$| + v$∙ A takes on the following form:
1 = ∙ !ä+ ä |∙ " + ∙
It is taken for granted, that here the sub-total d ∙ ä$|∙ A+ v$∙ A is equal to A.
Naturally these two forms are equivalent, because if we replace ä+ ä$|∙ A in the equation with 1-d ∙ ä, then we get
|ä
= ä+ ä |∙ = ä+ ä |∙ 1 − ∙ ä = ä+ ä |− ä |∙ ∙ ä
= ä+ ä |−1 −
1 − ∙ 1 − ∙ ä
= ä+ ä |− 1 − ∙ ä= ä |+ ∙ ä
Naturally, it is in theory also possible to guarantee the temporary life annuity, although this solution seems less justified than in case of a lifetime annuity.
Whatever the case, the equations are also extensions of the above equations in this case, i.e.
| ä
:|= ä |+ ä | :|
where g≤n, naturally. (We can see that if g=n then |$ä:| = ä$| and |ä:| = 0.) This restriction is not necessary in the case of a temporary life annuity:
|ä
:|= ä |+ ∙ ä:|
However, in this case it will also be true that:
|ä
:|= ä:+ ä |∙ :|
Because the annuity-certain is in fact a guaranteed annuity, the guaranteed period can only be properly interpreted in the case of life annuities. It is also possible to apply a guaranteed period to joint life annuities, but again, it is probably less relevant to those in view of the fact that the intention of the in-sured parties was intrinsically to leave their capital to the other party in the form of an annuity (we could in fact state that a single annuity with a guaran-teed period is ultimately a [poor] simulation of a joint life annuity). Neverthe-less, if it is also required for both temporary and joint annuities, then a guaran-teed period may naturally be added to the annuity according to the above.
There is a clear relationship between the orders of magnitude of the above-mentioned annuities. It is clear that:
ä|> ä:|> ä:|> ä*:|
and that
ä > ä
because (based on the relationships described in the first line above) in the case of ä| we are sure to receive payments for n years, but in case of ä:| for only a maximum of n years, in view of the fact that the insured individual currently aged x years could die sooner. If, however, the death of one out of two or more jointly insured individuals can also stop the flow of annuity payments, then the expected period of annuity payment will be even shorter in such cases. The same can be said in the case of lifelong single and joint life annuities.
This also means that the differences between the above-mentioned premi-ums will be higher than zero. Fortunately, their meanings are also easily inter-preted:
ä-ä: x receives EUR 1 per annum, but pays this back (negative annuity) while y is still alive, meaning x only receives payments after y has died. If x dies before y, no payment occurs at all. So, this is a conditional annuity: x begins receiving annuity payments once y has died. We can denote this with ä|or ä|= ä-ä.
In this case, our equation will change as follows:
1 = ∙ !ä+ ä|" +
ä| may also be regarded as a kind of asymmetric widow annuity. In this case, the financial situation of the two insured parties is different, and client y wish-es to assure an income for the other insured party, who is their dependent (cli-ent x) following their own death. The death of x does not affect y financially.
ä|-ä:|: the annuity is payable for those years of the n-year term during which x is no longer alive. If x survives to age x+n, then annuity payments do not
ä|-ä:|: the annuity is payable for those years of the n-year term during which x is no longer alive. If x survives to age x+n, then annuity payments do not