Annuity with Guarantee Period

→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛 advance, if the term is n years,

128

(10.63.) of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛

: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

128

(10.63.) of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛 It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If

128

(10.63.) of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛 10.64. is transformed to the following form:

128

(10.63.) of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛 The annuity in arrears versions are

128

(10.63.) of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛 of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛

of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛

of which the limit in case of n→∞, i.e. in case of lifetime annuity is the already discussed 1.

10.2.4. Certain annuities

Certain annuities cannot be considered insurance in a strict sense, since the fundamental feature of every insurance is that the benefits, or the degree of benefits provided by the insurer depend on the occurrence or non-occurrence of some random event. In case of this annuity, there is no such random event influencing the existence, the degree or the duration of the benefits paid by the insurer. Certain annuity means that for a speci-fied period, an annuity with specispeci-fied payment or a payment varying according to specispeci-fied rules will certainly be paid to the insured or the inheritor of the insured (beneficiary). It is important to talk about them all the same, because it can be an important complementing element of annuities and other types of insurance.

Most of the above categories can be applied also to the certain annuity. Accordingly, we can talk about certain annuities paid in advance or in arrears, immediate or deferred, temporary (paid for a certain period of time) and … not “lifetime” annuity– since the payment does not depend on the fact that the insured is alive or not –, but instead of this “infinite” annuity, which means that theoretically it is to be paid by the insurer forever to somebody (the current owner of the annuity).

We denote the net premium of the certain annuitywith a and ä, like the life annuities. The difference is clear, because in case annuity certains we do not note the age. The number in the right subscript means the term, emphasized by a “bend”. So

ä_𝑛𝑛|: is the single net premium of the certain annuity of 1 Forint yearly paid in advance, if the term isn years,

ä_∞|: is the single net premium of the certain infinite annuity of 1 Forint yearly paid in advance.

According to the above discussions it is easy to see that:

ä_𝑛𝑛|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1=1 − 𝑣𝑣^𝑛𝑛 1 − 𝑣𝑣

(10.64.) It can be proved that the infinite annuity is derived from the temporary annuity by increasing its term to infinity, choosing n to be infinite. If lim =0



→ n

n v . So, formula 10.64. is transformed to the following form:

ä_∞|= 𝑣𝑣⁰+ 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛−1+ ⋯ = 1 1 − 𝑣𝑣

(10.65.) The annuity in arrears versions are

𝑎𝑎_𝑛𝑛|= 𝑣𝑣¹+ 𝑣𝑣²+ ⋯ + 𝑣𝑣^𝑛𝑛= 𝑣𝑣 ∙1 − 𝑣𝑣^𝑛𝑛

Annuity with guarantee period is the connecting key on the one hand between lifetime annuities and certain annuities, and on the other hand between single life annuities and joint life annuities, which will be discussed in the next chapter. The guarantee period means that for a certain time period the insurer pays the annuity to the inheritor of the insured or another person declared by the policyholder, even if the insured dies during this period. So, we can say that the annuity with guarantee period is a joint life annuity concealed in the form of a single life annuity, or that the annuity with guarantee period is a certain type of “widows’ annuity” insurance (that will be discussed later).

The guarantee period can theoretically have two forms:

‚ guarantee period at the beginning: where the insurer guarantees that starting from

‚ guarantee period at the beginning: where the insurer guarantees that starting from

In document Life insurance (Pldal 186-189)