Joint Life Single Premium Insurance

Almost all types of insurances have a two or more person version, as we have seen in case of annuities.

Now, and in the further discussion we will only deal with two person insurances, and within these only the term insurance and the pure endowment insurance. We are on the opinion that, on the one hand the relations of the other two person insurances – if necessary – can be derived in an analogous way from the relations of single life insurances, and on the other hand from the practical point of view the relations of single life insurances are much more important, since currently mostly these dominate the market.

In case of two lives we simply regard as death the death of either of the two insured persons (i.e. the first death), and as living until maturity if both insured persons are alive at the end of the term.

In case of the two person term insurancewe try to think the following way: if all possible couples of agesxandytake out the policy ofn years term and 1 Forint sum assured, then the expected value of benefits paid by the insurer yearly will be:

(𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦− 𝑙𝑙_𝑥𝑥+1∙ 𝑙𝑙_𝑦𝑦+1);(𝑙𝑙_𝑥𝑥+1∙ 𝑙𝑙_𝑦𝑦+1− 𝑙𝑙_𝑥𝑥+2∙ 𝑙𝑙_𝑦𝑦+2); … ;(𝑙𝑙_{𝑥𝑥+𝑛𝑛−1}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛−1}− 𝑙𝑙_{𝑥𝑥+𝑛𝑛}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛}) (10.34.) If we denote the single premium in question by 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ _|̅, then the equivalence equation is:

(10.32.)

Here the interpretation of the parts of the equation is a little bit different. These are 1 year term special Term assurances with decreasing death sum assured. The speciality of them is, that somebody makes the contract at his/her x age, but they go into force after a 0, 1, …n-1 years “deferment period”. In another words, if the insured dies during the

“deferment period”, then the insurer does not pay any benefit payment.

The single premium pure endowment with premium refund is a pure endowment with death benefit. In case of the death of the insured, the insurer pays back the (gross) premiums paid by the policy holder so far – so in case of the single premium version, the single gross premium. From this angle this can also be considered as a special endowment policy. But in fact it is only a “little bit” insurance, it is almost the same as a savings account. The difference of the two, that in case of death the beneficiary does not get interest. That is why the policy can be considered as the “simulation” of a non-insurance product.

In the premium calculation – compare to the logic of the classical premium calculation described above – there is a “twist”, namely we need the gross premium already at the first step, when we calculate the net premium, but the calculation of the gross premium happens in the last step from the net premium.

If we note the gross single premium by AG, then we get the following equation for the single premium pure endowment with premium refund (which is a special endowment policy):

then we get the following premium formula:

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}=𝑛𝑛 𝑛𝑛 ∙

𝑀𝑀_𝑥𝑥− 𝑀𝑀_𝑥𝑥+1 𝐷𝐷_𝑥𝑥 +𝑛𝑛 − 1

𝑛𝑛 ∙

𝑀𝑀_𝑥𝑥+1− 𝑀𝑀_𝑥𝑥+2

𝐷𝐷_𝑥𝑥 + ⋯ +𝑛𝑛 − 1 𝑛𝑛 ∙

𝑀𝑀_{𝑥𝑥+𝑛𝑛−1}− 𝑀𝑀_𝑥𝑥+n 𝐷𝐷_𝑥𝑥

(10.32.) Here the interpretation of the parts of the equation is a little bit different. These are 1 year term special Term assurances with decreasing death sum assured. The speciality of them is, that somebody makes the cont-ract at his/her x age, but they go into force after a 0, 1, …n-1 years “deferment period”. In another words, if the insured dies during the “deferment period”, then the insurer does not pay any benefit payment.

The single premium pure endowment with premium refund is a pure endowment with death benefit. In case of the death of the insured, the insurer pays back the (gross) premiums paid by the policy holder so far – so in case of the single premium version, the single gross premium. From this angle this can also be considered as a special endowment policy. But in fact it is only a “little bit” insurance, it is almost the same as a savings account.

The difference of the two, that in case of death the beneficiary does not get interest. That is why the policy can be considered as the “simulation” of a non-insurance product.

In the premium calculation – compare to the logic of the classical premium calculation described above – there is a “twist”, namely we need the gross premium already at the first step, when we calculate the net premium, but the calculation of the gross premium happens in the last step from the net premium.

If we note the gross single premium by AG, then we get the following equation for the single premium pure endowment with premium refund (which is a special endowment policy):

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}= 𝐴𝐴𝐴𝐴_{𝑥𝑥:𝑛𝑛|}∙ 𝐴𝐴1_{𝑥𝑥:𝑛𝑛|}̅̅̅+ 𝐴𝐴_{𝑥𝑥:𝑛𝑛|} ¹

(10.33.) The classical premium calculation technique becomes uneasy at this insurance, because we have to an-ticipate, the difference of the gross and net premium. For this kind of calculation, the modern profit testing technique is more appropriate, because with the classical method it is hard to calculate the gross premium of this policy.

Further problem, that the pure endowment with premium refund is in almost every occasion a regular premium policy. But the death benefit of the regular premium pure endowment with premium refund will be different than the single premium version of it, so we can not use (10.33.). Therefore, the regular premium pure endowment with premium refund causes further problems in the logic of classical premium calculation, so later we will revisit later this topic.

10.1.4. Joint Life Single Premium Insurance

Almost all types of insurances have a two or more person version, as we have seen in case of annuities.

Now, and in the further discussion we will only deal with two person insurances, and within these only the term insurance and the pure endowment insurance. We are on the opinion that, on the one hand the relations of the other two person insurances – if necessary – can be derived in an analogous way from the relations of single life insurances, and on the other hand from the practical point of view the relations of single life insurances are much more important, since currently mostly these dominate the market.

In case of two lives we simply regard as death the death of either of the two insured persons (i.e. the first death), and as living until maturity if both insured persons are alive at the end of the term.

In case of the two person term insurancewe try to think the following way: if all possible couples of agesxandytake out the policy ofn years term and 1 Forint sum assured, then the expected value of benefits paid by the insurer yearly will be:

(𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦− 𝑙𝑙_𝑥𝑥+1∙ 𝑙𝑙_𝑦𝑦+1);(𝑙𝑙_𝑥𝑥+1∙ 𝑙𝑙_𝑦𝑦+1− 𝑙𝑙_𝑥𝑥+2∙ 𝑙𝑙_𝑦𝑦+2); … ;(𝑙𝑙_{𝑥𝑥+𝑛𝑛−1}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛−1}− 𝑙𝑙_{𝑥𝑥+𝑛𝑛}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛}) (10.34.) If we denote the single premium in question by 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ _|̅, then the equivalence equation is:

(10.33.) The classical premium calculation technique becomes uneasy at this insurance, because we have to anticipate, the difference of the gross and net premium. For this kind of calculation, the modern profit testing technique is more appropriate, because with the classical method it is hard to calculate the gross premium of this policy.

Further problem, that the pure endowment with premium refund is in almost every occasion a regular premium policy. But the death benefit of the regular premium pure endowment with premium refund will be different than the single premium version of it, so we can not use (10.33.). Therefore, the regular premium pure endowment with premium refund causes further problems in the logic of classical premium calculation, so later we will revisit later this topic.

10.1.4. Joint Life Single Premium Insurance

Almost all types of insurances have a two or more person version, as we have seen in case of annuities. Now, and in the further discussion we will only deal with two person insurances, and within these only the term insurance and the pure endowment

178 Banyár József: Life insurance

insurance. We are on the opinion that, on the one hand the relations of the other two person insurances – if necessary – can be derived in an analogous way from the relations of single life insurances, and on the other hand from the practical point of view the relations of single life insurances are much more important, since currently mostly these dominate the market.

In case of two lives we simply regard as death the death of either of the two insured persons (i.e. the first death), and as living until maturity if both insured persons are alive at the end of the term.

In case of the two person term insurance we try to think the following way: if all possible couples of ages x and y take out the policy of n years term and 1 Forint sum assured, then the expected value of benefits paid by the insurer yearly will be:

then we get the following premium formula:

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}=𝑛𝑛 Here the interpretation of the parts of the equation is a little bit different. These are 1 year term special Term assurances with decreasing death sum assured. The speciality of them is, that somebody makes the cont-ract at his/her x age, but they go into force after a 0, 1, …n-1 years “deferment period”. In another words, if the insured dies during the “deferment period”, then the insurer does not pay any benefit payment.

The single premium pure endowment with premium refund is a pure endowment with death benefit. In case of the death of the insured, the insurer pays back the (gross) premiums paid by the policy holder so far – so in case of the single premium version, the single gross premium. From this angle this can also be considered as a special endowment policy. But in fact it is only a “little bit” insurance, it is almost the same as a savings account.

The difference of the two, that in case of death the beneficiary does not get interest. That is why the policy can be considered as the “simulation” of a non-insurance product.

In the premium calculation – compare to the logic of the classical premium calculation described above – there is a “twist”, namely we need the gross premium already at the first step, when we calculate the net premium, but the calculation of the gross premium happens in the last step from the net premium.

If we note the gross single premium by AG, then we get the following equation for the single premium pure endowment with premium refund (which is a special endowment policy):

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}= 𝐴𝐴𝐴𝐴_{𝑥𝑥:𝑛𝑛|}∙ 𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹

(10.33.) The classical premium calculation technique becomes uneasy at this insurance, because we have to an-ticipate, the difference of the gross and net premium. For this kind of calculation, the modern profit testing technique is more appropriate, because with the classical method it is hard to calculate the gross premium of this policy.

Further problem, that the pure endowment with premium refund is in almost every occasion a regular premium policy. But the death benefit of the regular premium pure endowment with premium refund will be different than the single premium version of it, so we can not use (10.33.). Therefore, the regular premium pure endowment with premium refund causes further problems in the logic of classical premium calculation, so later we will revisit later this topic.

10.1.4. Joint Life Single Premium Insurance

Almost all types of insurances have a two or more person version, as we have seen in case of annuities.

Now, and in the further discussion we will only deal with two person insurances, and within these only the term insurance and the pure endowment insurance. We are on the opinion that, on the one hand the relations of the other two person insurances – if necessary – can be derived in an analogous way from the relations of single life insurances, and on the other hand from the practical point of view the relations of single life insurances are much more important, since currently mostly these dominate the market.

In case of two lives we simply regard as death the death of either of the two insured persons (i.e. the first death), and as living until maturity if both insured persons are alive at the end of the term.

In case of the two person term insurancewe try to think the following way: if all possible couples of agesxandytake out the policy ofn years term and 1 Forint sum assured, then the expected value of benefits paid by the insurer yearly will be:

(𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦− 𝑙𝑙𝑥𝑥+1∙ 𝑙𝑙𝑦𝑦+1);(𝑙𝑙𝑥𝑥+1∙ 𝑙𝑙𝑦𝑦+1− 𝑙𝑙𝑥𝑥+2∙ 𝑙𝑙𝑦𝑦+2); … ;(𝑙𝑙𝑥𝑥+𝑛𝑛−1∙ 𝑙𝑙𝑦𝑦+𝑛𝑛−1− 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛) (10.34.) If we denote the single premium in question by 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ _|̅, then the equivalence equation is:

(10.34.) If we denote the single premium in question by

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}=𝑛𝑛 Here the interpretation of the parts of the equation is a little bit different. These are 1 year term special Term assurances with decreasing death sum assured. The speciality of them is, that somebody makes the cont-ract at his/her x age, but they go into force after a 0, 1, …n-1 years “deferment period”. In another words, if the insured dies during the “deferment period”, then the insurer does not pay any benefit payment.

The single premium pure endowment with premium refund is a pure endowment with death benefit. In case of the death of the insured, the insurer pays back the (gross) premiums paid by the policy holder so far – so in case of the single premium version, the single gross premium. From this angle this can also be considered as a special endowment policy. But in fact it is only a “little bit” insurance, it is almost the same as a savings account.

The difference of the two, that in case of death the beneficiary does not get interest. That is why the policy can be considered as the “simulation” of a non-insurance product.

In the premium calculation – compare to the logic of the classical premium calculation described above – there is a “twist”, namely we need the gross premium already at the first step, when we calculate the net premium, but the calculation of the gross premium happens in the last step from the net premium.

If we note the gross single premium by AG, then we get the following equation for the single premium pure endowment with premium refund (which is a special endowment policy):

𝐴𝐴_{𝑥𝑥:𝑛𝑛|}= 𝐴𝐴𝐴𝐴_{𝑥𝑥:𝑛𝑛|}∙ 𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹

(10.33.) The classical premium calculation technique becomes uneasy at this insurance, because we have to an-ticipate, the difference of the gross and net premium. For this kind of calculation, the modern profit testing technique is more appropriate, because with the classical method it is hard to calculate the gross premium of this policy.

Further problem, that the pure endowment with premium refund is in almost every occasion a regular premium policy. But the death benefit of the regular premium pure endowment with premium refund will be different than the single premium version of it, so we can not use (10.33.). Therefore, the regular premium pure endowment with premium refund causes further problems in the logic of classical premium calculation, so later we will revisit later this topic.

10.1.4. Joint Life Single Premium Insurance

Almost all types of insurances have a two or more person version, as we have seen in case of annuities.

Now, and in the further discussion we will only deal with two person insurances, and within these only the term insurance and the pure endowment insurance. We are on the opinion that, on the one hand the relations of the other two person insurances – if necessary – can be derived in an analogous way from the relations of single life insurances, and on the other hand from the practical point of view the relations of single life insurances are much more important, since currently mostly these dominate the market.

In case of two lives we simply regard as death the death of either of the two insured persons (i.e. the first death), and as living until maturity if both insured persons are alive at the end of the term.

In case of the two person term insurancewe try to think the following way: if all possible couples of agesxandytake out the policy ofn years term and 1 Forint sum assured, then the expected value of benefits paid by the insurer yearly will be:

(𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦− 𝑙𝑙𝑥𝑥+1∙ 𝑙𝑙𝑦𝑦+1);(𝑙𝑙𝑥𝑥+1∙ 𝑙𝑙𝑦𝑦+1− 𝑙𝑙𝑥𝑥+2∙ 𝑙𝑙𝑦𝑦+2); … ;(𝑙𝑙𝑥𝑥+𝑛𝑛−1∙ 𝑙𝑙𝑦𝑦+𝑛𝑛−1− 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛) (10.34.) If we denote the single premium in question by 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ _|̅, then the equivalence equation is:, then the equivalence equation is:

123 Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷_{𝑥𝑥𝑦𝑦} = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛 Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ ̅_|= v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}¹ ̅_|=v ∙(𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷_{𝑥𝑥𝑦𝑦} = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|} (10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛}̅¹_|= 𝑣𝑣ⁿ∙ 𝑙𝑙_{𝑥𝑥+𝑛𝑛}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛} Multiplying by v^x, and using the two person commutation numbers, the already (in single life version) introduced D_xy and (the not yet introduced) N_xy introduced earlier we get: Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introduced Dxy and (the not yet introduced) Nxy introduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷𝑥𝑥𝑦𝑦 = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛

179 Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedD^xyand (the not yet introduced) N^xyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}¹ _̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where 𝐷𝐷_{𝑥𝑥𝑦𝑦}= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷𝑥𝑥𝑦𝑦 = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙_{𝑥𝑥+𝑛𝑛}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛} Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷_{𝑥𝑥𝑦𝑦} = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛 From this (by anticipating the premium of annuities):

123 Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷𝑥𝑥𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷_{𝑥𝑥𝑦𝑦}= 𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷𝑥𝑥𝑦𝑦 = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛 The equivalence equation of the two person pure endowment insurance is:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = 𝑣𝑣¹∙(𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦− 𝑙𝑙𝑥𝑥+1∙ 𝑙𝑙𝑦𝑦+1)+ Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷𝑥𝑥𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷𝑥𝑥𝑦𝑦 = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙_𝑥𝑥∙ 𝑙𝑙_𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|} 1̅ = 𝑣𝑣ⁿ∙ 𝑙𝑙_{𝑥𝑥+𝑛𝑛}∙ 𝑙𝑙_{𝑦𝑦+𝑛𝑛} Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁𝑥𝑥𝑦𝑦− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷_{𝑥𝑥𝑦𝑦} = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛 Multiplying byv^x, and using the two person commutation numbers, the already (in single life version) introducedDxyand (the not yet introduced) Nxyintroduced earlier we get:

𝐷𝐷_{𝑥𝑥𝑦𝑦}∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}¹ _̅ = v ∙(𝐷𝐷_{𝑥𝑥𝑦𝑦}+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1))−(𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ 𝐷𝐷(𝑥𝑥+2)(𝑦𝑦+2)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))=

= v ∙ (𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛)) − (𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

(10.37.) Where𝐷𝐷𝑥𝑥𝑦𝑦= 𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝑣𝑣^𝑥𝑥, and 𝑁𝑁_{𝑥𝑥𝑦𝑦}= 𝐷𝐷𝑥𝑥𝑦𝑦+ 𝐷𝐷(𝑥𝑥+1)(𝑦𝑦+1)+ ⋯ + 𝐷𝐷(𝑥𝑥+𝑛𝑛−1)(𝑦𝑦+𝑛𝑛−1)+ ⋯

From this (by anticipating the premium of annuities):

𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}1 ̅ =v ∙(𝑁𝑁_{𝑥𝑥𝑦𝑦}− 𝑁𝑁(𝑥𝑥+𝑛𝑛)(𝑦𝑦+𝑛𝑛))−(𝑁𝑁(𝑥𝑥+1)(𝑦𝑦+1)− 𝑁𝑁(𝑥𝑥+𝑛𝑛+1)(𝑦𝑦+𝑛𝑛+1))

𝐷𝐷_{𝑥𝑥𝑦𝑦} = v ∙ ä_{𝑥𝑥𝑦𝑦:𝑛𝑛|}− 𝑎𝑎_{𝑥𝑥𝑦𝑦:𝑛𝑛|}

(10.38.) The equivalence equation of thetwo person pure endowment insuranceis:

𝑙𝑙𝑥𝑥∙ 𝑙𝑙𝑦𝑦∙ 𝐴𝐴_{𝑥𝑥𝑦𝑦:𝑛𝑛|}̅1 = 𝑣𝑣ⁿ∙ 𝑙𝑙𝑥𝑥+𝑛𝑛∙ 𝑙𝑙𝑦𝑦+𝑛𝑛

In document Life insurance (Pldal 177-180)