CALCULATION OF PREMIUMS. We are now pre pared to investigate the method of determining net premium rates from the mortality table. It will be most convenient to start with the sim plest form of policy and to proceed to the more complex forms. It a person at age twenty-five desires to insure his life for $1.1100 for one year, what premium ought lie to pay? By consulting the mortality table we find that of 811,835 persons living at age twenty-five, 69S die within a year. The probability of death is therefore represented by the fraction The risk involved in in suring his life for $10011 for one year is there fore $1000 X or $7.77. But the premium is pai-I at the beginning of the year, the in demnity at the end. It is therefore necessary to allow for the interest, say at 4 per cent., which the premium will earn during the year. This is done by discounting the $7.77 for one year at 4 per cent. Doing this we find the net premium to be $7.47. If the insurance is renewed at the close of the year the new premium is based on the prob ability of the death of the person at age twenty six. and amounts to $7.5S; and with every suc ceeding year the premimn increases according to the increased probability of death during the year. This method of paying for insurance is known as the natural premium plan.
The premium necessary to secure insurance for a number of years by a single payment at the beginning of the first year is determined in a similar manner. Thus to find the single premium for two years' insurance, it would he necessary to discount the risk I that is, the amonnt of in surance mulitplied by the probability of death) for the first year at 4 per cent. simple discount for one year. and the risk for the second year at 4 per cent. compound discount for two years. The sum of the two results is the single advanced premium for two insurance. continu ing the process the single premium for an insur ance to run any stated number of years or even for life may be ascertained. insurance limited to a :stated number of years is known as term insurance.
In the great majority of cases term insurance is paid for neither by a single premium at the be ginning of the term nor by natural premiums ad vancing from year to year. but by a uniform annual premium during the term of insurance. Such a premium is known as a 'level' annual premium. Let us see how it would be. determined for a two years' term. The two annual pre miums must evidently have the same value to the company as the single advance premium. with
proper allowance for the possibility that the in sured will not be alive to pay the second pre mium. It is necessary. therefore. to find out the present value of $1 to be paid at once and $1 paid a year from now if the insured is living at that date. To ascertain the latter amount we must first discount the dollar for one year at 4 per cent.. and then multiply the result by the fraction whose numerator is the number living at the end of the year and whose denominator is the number living, now. That fraction represent; the probability that a person living now will be living a year from now: and the result of the whole process is the present value of the $1 pay able a year from now, subject to the contingency of death during the year. If this amount is added to the dollar payable at the beginning of the first year the sum is the present calve of the promise of tine insured to pay a dollar in ad vance for two •cars if living. To find how many dollars in a level annual premium are equal in value to a single premium payable in advance, divide the single premium by the present value of $1 if living. The mathematical operation of finding the level annual premium fur :stun() in surance for two years at age twenty-tive is as follows: Risk for first year, $7.77, discounted nt 4 hwr rent.
Lo one year . 17 47 Risk for see, nal year, $7.sa7, discom t•d at 4 per for two years 7 29 Single :el va me premium for two years' insura TIC,. ;14 76 Present value $1 payable now . 1 oU Present value of $1 payable one :Lon s9137 year from now, subject to the- X 95 contingency of death Sl.14 ste,37, Present valm of annual premium of $1 for two years $1 Level al1111111i premium, .w 7b7 To find the level premimn for three rears' insurance. add to the $14.76 the risk of the third year discounted for three years at compound dis count, and to the $1.95 the present worth of the promise to pay $1 two years from now, and divide as before. To find that present worth. multiply the present value of $1 at 4 per cent. compound discount for two years by the fraction whose numerator is the number living at the beginning of the third year and whose' denomina tor is the number living now. By continuing this process the net level annual premium for term insurance for any number of years will be found.