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Add to this the net single premium for an insurance of twenty instalments certain to obtain the entire net single premium.

Continuous Instalment Provisions : Of a much simpler character is the computation of the amount of instalment for twenty years certain and for the after-lifetime of the beneficiary, in lieu of a lump payment of $i,000, for in stance, the amount of the instalment to be determined by the age of the beneficiary at the time of the death of the insured. Such provisions, with a table of such values per $1,000 insured, are frequently found in policies nowadays.

First find the present value of twenty instalments of $i each, payable annually in advance, and add the pres ent value of a life annuity of $i at the present age of the beneficiary, less the present value of a life annuity at the same age limited to 20 years. Since this is the pres ent value of an instalment of $1, and the sum of $i,000 is to be employed to purchase the instalment benefits, divide $i,000 by this present value; the quotient will be the amount of continuous annual instalment which can be paid in lieu of a lump sum of $z,000 if the insured die this year that is, while the beneficiary is at her pres ent age.

The amount of the instalment if the insured dies the second, third, fourth, etc., year is computed in a similar manner. If desired, the amounts for each age may be computed and a table of such amounts may be endorsed upon the policy.

Income or Guaranteed Interest Bonds : If a company makes its computations at 3 per cent. interest, for instance, and is really willing to guarantee that rate of interest in . 69 definitely, it can, if it choose, without any addition to the net premium, permit the proceeds of an endowment to remain with it, paying interest at 3 per cent. annually thereon during a life or lives such as the life of the insured or the lives of the insured and the beneficiary or for a definite term of years, and paying over the prin cipal sum upon the termination of the life or lives or of the agreed period. In like manner, upon the death of the insured it could hold the sum insured, paying inter est at such rate to the beneficiary during her lifetime • and upon her death paying over the principal sum. Poli cies with such provisions i. e., for the payment of in terest at the rate assumed by the company in computing its premiums and reserves are sometimes issued with out increase of premium.

Frequently, however, the income or guaranteed "in terest," so called, on the nominal amount of the bond or insurance usually $I,000 is higher than the company really counts upon earning. Then, in order to find the net single premium, the present value, at the moment of the death of the insured, of all the sums actually to be paid, must be computed and be taken as the true and full amount of the insurance.

Suppose the bond provides that, beginning at the death of the insured or also at the policy's maturity, if en dowment insurance $5o shall be paid at the end of each year for 20 years, and then $i,000. Suppose the com pany really counts upon earning only 3 per cent. interest. The present value, at the death of the insured (or on maturity, if an endowment), of all the payments to be made is, at 3 per cent. annually compounded, present value of principal, $553.67; of "interest," $743.88. Total, $743.88 $553.67 = $1,297.55. Compute, then, in the ordinary manner, the net single premium for an insur ance or an endowment insurance, as the case may be, of 70 $1,297.55, and you have the net single premium for this bond. The effect is that the bond is really sold at a pre mium over par of about 3o per cent.

The same result may be arrived at in another manner, as follows : The nominal principal sum of $1,000 pro duces $30 a year income at 3 per cent. interest, leaving $2o a year for 20 years to be provided for. The dis counted value of $2o a year is, at 3 per cent. interest, an nually compounded, $297.55, which sum, being added to $i,000, produces $1,297.55 as the actual principal sum insured.

A special form of this bond is to pay $1,000 immedi ately upon death or maturity, $5o a year at the end of each year for 20 years, and then $1,000. This is, of course, worth just $1,000 more i. e., it is an insurance, on a 3 per cent. basis, for $2,297.55.

There remains the case of a policy which pays $5o at the end of each year for the after-lifetime of the bene ficiary, if surviving the insured, and $1,000 at the death of the beneficiary. The interest which $1,000 would actually yield at 3 per cent. is $30 a year; provision, there fore, must be made for $20 a year additional for the re maining lifetime of the beneficiary after the death of the insured. Remembering that the present value of a sur vivorship or reversionary annuity is the present value of an annuity for the life of the beneficiary, less the pres ent value of an annuity for the joint lives of the insured and the beneficiary, we have: The entire net single premium is the net single pre mium for an insurance of $1,000 on the life of the in sured (which, as has been said, can be treated as produc ing $30 interest per annum during the beneficiary's after lifetime and being still worth $1,000 at the time of her decease) plus the present value of an annuity of $20 on the life of the beneficiary, less the present value of 71 an annuity of $2o on the joint lives of the insured and the beneficiary. This would yield $1,030, however, at the death of the beneficiary, as a year's interest woulcl be earned, and so would need to be slightly modified in practice.

All of the foregoing are on the basis that the payments are at the end of the year; in practice they are usually at the beginning or in instalments throughout the year, in which cases suitable adjustments must be made.

Annual Premiums : The net annual premium for any of these insurances, if involving one life only, is com puted by dividing the net single premium or present value by the present value of a life annuity due of $1 for the life of the insured when premiums are payable through out life, or the present value of the corresponding tem porary life annuity due if premiums are payable for a lim ited term only.

Where the insurance involves survivorship, it is usually assumed that payment of premiums will not continue be yond the lifetime of either the insured or the beneficiary, and the net single premium is divided by the present value of a joint-life annuity due or a temporary joint-life an nuity due, as the case may require.

If only a part involves survivorship, that part of the entire net single premium should be divided by a joint life annuity due and the remainder by the annuity due for the life of the insured; and in such case the part of the annual premium arrived at by dividing by the joint-life annuity is not required after the beneficiary's death, if it takes place before the death of the insured.

Thus, a continuous instalment policy becomes a mere insurance of twenty instalments certain when the bene ficiary dies before the insured, and, therefore, from that time forth the net annual premium should be only the net annual premium for an insurance of twenty instal ments certain.