The Net Premium of Regular Premium Payment Insurance

In case of the regular premium insurance the premium is paid not all at once, at the beginning of the term, but spread through the whole insurance term, in instalments. Most life insurances have both a single premium and a regular premium version (an exception

195

is e.g. the immediate annuity, which doesn’t have a regular premium version¹⁰¹, and the term fix insurance, where the single premium version would be problematic¹⁰²).

To keep the matter simple, we suppose that the single premium is paid at the beginning of each insurance year, in equal payments, except in the sub-chapter discussing the specialities of premium frequencies other than annual. This annual premium is derived from the single premium of the insurance with the same parameters, but the single premium version. Obviously, this cannot be done simply by dividing the single premium with the number of years in the term to get the annual premium. This has two causes:

Compared to the single premium insurance, the insurer suffers interest loss in case of the regular premium insurance, since the greater part of the premium is received only years later, and, until then, the insurer doesn’t earn interest on these parts.

The insurer receives the total single premium. But in case of the regular premium, the insurer cannot be totally certain about receiving all of the premium payments, because if the insured dies during the term, then further premium payment ceases.

Due to these causes, the annual premium will be higher than the single premium divided by the number of years in the term.

As we have indicated, the period of premium payment can be equal to the insurance term, but it can also be shorter. If m denotes the number of years of premium payment, and the usual n denotes the insurance term, then it is always true that:

dies before the death of the first insured, the insurance is terminated without any benefit payment. The single premium of this insurance is:

ä_𝑦𝑦− ä_{𝑥𝑥𝑦𝑦}

(10.101.) The meaning of the formula is: the secondary insured receives a yearly 1 Forint annuity starting from the commencement of the insurance, but until the primary insured is also alive (i.e. both are alive), they pay this 1 Forint yearly annuity back to the insurer.

Naturally a number of other special annuity types can be imagined beside the above discussed ones.

10.3. T^HEN^ETP^{REMIUM OF}R^EGULAR P^REMIUMP^AYMENTI^NSURANCE

In case of the regular premium insurance the premium is paid not all at once, at the beginning of the term, but spread through the whole insurance term, in instalments. Most life insurances have both a single pre-mium and a regular prepre-mium version (an exception is e.g. the immediate annuity, which doesn’t have a regular premium version¹⁰¹, and the term fix insurance, where the single premium version would be problematic¹⁰²).

To keep the matter simple, we suppose that the single premium is paid at the beginning of each in-surance year, in equal payments, except in the sub-chapter discussing the specialities of premium frequencies other than annual. This annual premium is derived from the single premium of the insurance with the same parameters, but the single premium version. Obviously, this cannot be done simply by dividing the single pre-mium with the number of years in the term to get the annual prepre-mium. This has two causes:

Compared to the single premium insurance, the insurer suffersinterest lossin case of the regular pre-mium insurance, since the greater part of the prepre-mium is received only years later, and, until then, the insurer doesn’t earn interest on these parts.

The insurer receives the total single premium. But in case of the regular premium, the insurer cannot be totally certain about receiving all of the premium payments, because if theinsured dies during the term, then further premium payment ceases.

Due to these causes, the annual premium will be higher than the single premium divided by the number of years in the term.

As we have indicated, the period of premium payment can be equal to the insurance term, but it can also be shorter. If mdenotes the number of years of premium payment, and the usualndenotes the insurance term, then it is always true that:

𝒎𝒎 ≤ 𝒏𝒏

The reason of this the practical insurance principle according to “the premium always goes to the insurer in advance”. Namely, the insurer can enforce premium payment from the policy holder by refusing paying the insurance benefits in case of arrears of premiums. If the insurer gave the benefits in advance (e.g. if the premium payment period exceeds the term), then it would loose this simple but effective tool, and would be able to en-force the premium payment only by expensive methods which made the insurance premiums disproportionately high.

Hereunder we suppose that the premium payment period equal to the term. The permiums of cases different from this could be derived simply from this case.

In order to be able to derive the annual premium, we have to realise that the annual premium is tech-nically the same as a temporary annuity paid in advance, where the annuity payment is the annual premium, and the term of the annuity equals the premium payment term, only the annuity payment is not paid by the insurer to the insured, but vice versa. This last circumstance does not influence the value of the annuity.

Starting out from the above consideration, the equivalence equation can be written the following way in case of all regular premium insurances, if the annual premium is P:

101Although even this can be imagined if the policyholder and the insured are not the same. Then the policyholder pays the premium in instalments while the insured already receives annuity payments. This construction was used e.g. in Hungary in the 1990s, when the disability pension liability of closed mines was transferred to insurers. On the other hand, these are usually individual annuity constructions, such products are not developed, because people typically take out annui-ties for themselves. Of course a greater market demand can be imagined, in which case these products would appear!

102We will discuss the term fix insurance under 10.1.3.

The reason of this the practical insurance principle according to “the premium always goes to the insurer in advance”. Namely, the insurer can enforce premium payment from the policy holder by refusing paying the insurance benefits in case of arrears of premiums. If the insurer gave the benefits in advance (e.g. if the premium payment period exceeds the term), then it would loose this simple but effective tool, and would be able to enforce the premium payment only by expensive methods which made the insurance premiums disproportionately high.

Hereunder we suppose that the premium payment period equal to the term. The permiums of cases different from this could be derived simply from this case.

In order to be able to derive the annual premium, we have to realise that the annual premium is technically the same as a temporary annuity paid in advance, where the

101 Although even this can be imagined if the policyholder and the insured are not the same. Then the policyholder pays the premium in instalments while the insured already receives annuity payments.

This construction was used e.g. in Hungary in the 1990s, when the disability pension liability of closed mines was transferred to insurers. On the other hand, these are usually individual annuity constructions, such products are not developed, because people typically take out annuities for themselves. Of course a greater market demand can be imagined, in which case these products would appear!

102 We will discuss the term fix insurance under 10.1.3.

annuity payment is the annual premium, and the term of the annuity equals the premium payment term, only the annuity payment is not paid by the insurer to the insured, but vice versa. This last circumstance does not influence the value of the annuity.

Starting out from the above consideration, the equivalence equation can be written the following way in case of all regular premium insurances, if the annual premium is P:

137

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃𝑥𝑥=𝐴𝐴_𝑥𝑥 ä_𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it is ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴1_{𝑥𝑥:𝑛𝑛|}̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃_𝑥𝑥=𝐴𝐴_𝑥𝑥 ä𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the“staged” term insurance:

. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

137

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃_𝑥𝑥=𝐴𝐴𝑥𝑥

ä𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insurance is:

137

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃𝑥𝑥=𝐴𝐴_𝑥𝑥 ä_𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.104.) The regular premium of the pure endowment insurance is:

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}1̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|} ¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|} ¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃𝑥𝑥=𝐴𝐴𝑥𝑥

ä_𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.105.) The regular net premium of the endowment insurance is:

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴1_{𝑥𝑥:𝑛𝑛|}̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insurance is:

𝑃𝑃𝑥𝑥:𝑛𝑛|̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|} ¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃_𝑥𝑥=𝐴𝐴𝑥𝑥

ä𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.106.) The regular net premium of the whole life insurance:

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅=𝐴𝐴1_{𝑥𝑥:𝑛𝑛|}̅̅̅

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃_𝑥𝑥=𝐴𝐴_𝑥𝑥 ä𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x|¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

197

ä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃 = 𝐴𝐴

(10.102.) since the expected income of the insurer is exactly the same as the value of the annuity paid to the insurer from the client, i.e. it isä_{𝑥𝑥:𝑛𝑛|}∙ 𝑃𝑃. The expected value of payout is the same as in case of the single premium insurance, since the single and regular payment versions do not differ in this respect. This way:

𝑃𝑃 = 𝐴𝐴 ä_{𝑥𝑥:𝑛𝑛|}

(10.103.) Let’s apply this general relation to concrete insurances! We are on the opinion that having derived the formulae of the single premiums now it is enough in most cases to just write the concrete formulae.

10.3.1. The Regular Net Premium of the Single Life Insurances The regular premium of the term insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹_̅̅̅=𝐴𝐴¹_{𝑥𝑥:𝑛𝑛|̅̅̅}

ä_{𝑥𝑥:𝑛𝑛|}

(10.104.) The regular premium of the pure endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹=𝐴𝐴_{𝑥𝑥:𝑛𝑛|}¹ ä_{𝑥𝑥:𝑛𝑛|}

(10.105.) The regular net premium of the endowment insuranceis:

𝑃𝑃_{𝑥𝑥:𝑛𝑛|}̅̅̅= 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}1̅̅̅+ 𝑃𝑃_{𝑥𝑥:𝑛𝑛|}¹

(10.106.) The regular net premium of the whole lifeinsurance:

𝑃𝑃𝑥𝑥=𝐴𝐴_𝑥𝑥 ä_𝑥𝑥

(10.107.) - at least theoretically. In practice the whole life insurance is calculated quite often as an endowment until a very high age (e.g. 85), so its premium is:

P_x:85−x|̅̅̅̅̅̅̅̅= P_x:85−x|1̅̅̅̅̅̅̅̅+ P_x:85−x| ¹

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

(10.108.) Naturally, this is not a real whole life insurance, but it is called this way.

The net regular premum of the “staged” term insurance:

138

P_x:n|̅ =

n ∙ Mx− (Rx+1− Rx+n+1) n ∙ D_x

ä_x:n|

It is worth to note, that this premium quite often leads to a negative reserve, which has to be avoided.

That is why in the case of this insurance the premium payment period is quite often shortened – not because of marketing considerations, but because it is a kind of professional necessity. This topic is treated in detail at reser-ves.

The Net Regular Premium of the Term Fix Insurance:

The Net Regular Premium of the Term Fix Insurance:

In document Life insurance (Pldal 195-198)