of 'loading' to cover the expeiises of the insurance company, and to provide for contingencies. The loading is usually from 10 to 40 per cent, of the premium.
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. If a person at age twenty-five desires to insure his life for $1000 for one year,
what premium ought he to pay '! By consulting the mortality table we find that of 89,835 persons living at age twenty-five, 698 die within a year. The probability of death is therefore represented by the fraction 698⁄89385. The risk involved in in- suring his life for $1000 for one year is there fore $1000 × 698⁄89385, or $7.77. But the premium is paid at the' beginning of the year, the in- demmty 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. It the insurance is renewed at the close of the year the new premium is based on the probability ability of the death of the person at age twenty- six, and amounts to $7.58; and with every suc- ceeding year the premium 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 be necessary
to discount the risk (that is, the amount 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 years insurance. By 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 xases 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 represents
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 value of the
promise of the insured to pay a dollar in ad-
ance for two years 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 for $1000 in-
surance for two years at age twenty-five is as
follows:
Risk for first year, $7.77, discounted at 4 per cent. for one year $7 47 Risk for scond year, $7,887, discounted at 4 percent. for two years 7 29
Single advance premium for two years' insurance... $14 76 Present value of $1 payable now $1.00 Present value of $1 payable one year from now, subject to the contingency of death
$1.00 x 81937 $1.04 80835
or 95
Present value of annual premium of $1 for two years $1 95 Level annual premium. $14.76-r $1.95 or 7 57
To find the level premium for three years' 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 conitingent 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 diswcount 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,
In the case of insurance to run for life there is apparently a new element introduced, since indemnity is certain sooner or later to become a claim. But the difference is only in appear- ance. If the Actuaries' table be used. a whole- life policy taken out at twenty-five may be treated as a term policy for seventy-four years, since the tbale assumes the death of the last survivor at ninety-nine. In calculating the risk for the seventy-fourth year the fraction represent- ing the probability of death becomes unity, and the risk assumed for that year equals the face of the policy. This risk would then be dis- dicounted at 4 per cent, compound discount for seventy-four years, and the result added to re- sults similarly obtained for previous years. On the other hand, the promise to pay .$1 each year for life ceas to have any value for any uear after the seventy-fourth, since on the sev- pnty-fifth year the numerator of the fraction rep- resenting the probability of survival becomes zero.
Endowment insurance to-day is quite as much in evidence as pure life insurance. An endow- ment is a promise to pay a person a stated sum at a future day provided the person is living at that day. Endowment insurance usually provides also for the payment of an insurance indemnity if the insured dies before the end of the en- dowment period. The two transactions, however, the insurance and the endowment, are entirely distinct and could perfectly well be carried on by different companies. In the case of endow- ment insurance the premium contains two ele- ments. one to pay for the endowment, the other to pay for the insurance. Let us consider how each is determined.
